Explanation:
Okay, so I need to find the angle between vectors a and a - b. The problem gives me the angle between a and b, which is 29 degrees, and the angle between b and a - b, which is 84 degrees. Hmm, let me try to visualize this. Maybe I can draw a diagram to help me understand the relationships between these vectors.

First, let me recall that the angle between two vectors can be found using the dot product formula. The formula is:

cosθ = (a · b) / (|a||b|)

But I don't know the magnitudes of the vectors a or b. Maybe I can assign some variables to them? Let me denote |a| as A and |b| as B. Then, the angle between a and b is 29 degrees, so:

cos(29°) = (a · b) / (AB)

Similarly, the angle between b and a - b is 84 degrees. Let me write that down:

cos(84°) = [b · (a - b)] / (|b||a - b|)

Let me compute the numerator of that second equation. The dot product b · (a - b) is equal to b · a - b · b, which is (a · b) - |b|². So:

cos(84°) = [ (a · b) - B² ] / (B |a - b| )

Hmm, but I don't know |a - b| either. Maybe I can find |a - b| using the law of cosines? Since we have vectors a and b with an angle between them, the magnitude of a - b can be found by:

|a - b|² = |a|² + |b|² - 2|a||b|cosθ, where θ is the angle between a and b. So:

|a - b|² = A² + B² - 2AB cos(29°)

Therefore, |a - b| = sqrt(A² + B² - 2AB cos(29°))

But this seems complicated because I don't know A or B. Maybe there's another way. Let me think.

Alternatively, maybe I can consider the triangle formed by vectors a, b, and a - b. Wait, if I think of vectors a and b as two sides of a triangle, then a - b would be the third side. But actually, the triangle would be formed by vectors a, b, and a - b? Wait, no. If you have vectors a and b, then a - b is equivalent to a + (-b). So, if you place the tail of vector b at the tip of vector a, then a - b would be the vector from the tail of a to the tip of -b. Maybe I need to draw this.

Alternatively, maybe I can use the law of sines or cosines in the triangle formed by vectors a, b, and a - b. Let me try that approach.

Let me denote the angle between a and a - b as θ, which is what we need to find. The angle between b and a - b is given as 84 degrees. The angle between a and b is 29 degrees. So, if I imagine a triangle with sides corresponding to vectors a, b, and a - b, then the angles opposite these sides would be related. Wait, but the angles between the vectors are not necessarily the angles of the triangle. Wait, actually, if we consider the triangle formed by vectors a, b, and a - b, then the angles of the triangle would be related to the angles between the vectors. Let me think.

Let me consider the triangle where one side is vector a, another side is vector b, and the third side is vector a - b. Wait, but in reality, the triangle would have sides of length |a|, |b|, and |a - b|, and the angles between these sides would correspond to the angles between the vectors. However, the angle between a and b is 29 degrees, which would be the angle between sides a and b in the triangle. Then, the angle between b and a - b is 84 degrees, which would be the angle between side b and side a - b in the triangle. Then, the angle between a and a - b is θ, which is the angle between side a and side a - b in the triangle. Therefore, the three angles of the triangle would be 29°, 84°, and θ. Wait, but the sum of angles in a triangle is 180°, so:

29° + 84° + θ = 180°

Therefore, θ = 180° - 29° - 84° = 180° - 113° = 67°

Wait, is that possible? If that's the case, then the angle between a and a - b is 67 degrees? But wait, is this correct? Because the angles between the vectors are not necessarily the angles of the triangle. Wait, let me think again.

Suppose we have vectors a and b with an angle of 29° between them. Then, if we form a triangle with sides a, b, and a - b, then the angle between a and b in the triangle is indeed 29°, because that's the angle between the vectors. However, the angle between b and a - b is given as 84°, which would be the angle between side b and side a - b in the triangle. Similarly, the angle between a and a - b is θ, which is the angle between side a and side a - b in the triangle. Therefore, the three angles of the triangle would be 29°, 84°, and θ, which sum to 180°, so θ = 67°. Therefore, the answer is 67 degrees?

Wait, but this seems too straightforward. Is there a mistake here? Because I might be confusing the angles between vectors with the angles of the triangle. Let me verify.

Let me consider the triangle formed by vectors a, b, and a - b. Wait, actually, if you have vectors a and b, then the vector a - b is the vector from the tip of b to the tip of a if they are placed tail to tail. Wait, maybe I need to think of the triangle as follows: if you place the vectors a and b tail to tail, then the vector a - b would be the vector from the tip of b to the tip of a. Therefore, the triangle would have sides a, b, and a - b, with the angle between a and b being 29°, the angle between b and a - b being 84°, and the angle between a and a - b being θ. Therefore, the three angles of the triangle would be 29°, 84°, and θ, which sum to 180°, so θ = 67°. Therefore, the answer is 67 degrees. But let me check if this is correct with another method to be sure.

Alternatively, let me use the law of sines. In the triangle, the sides are |a|, |b|, and |a - b|. The angles opposite these sides would be the angles opposite to them. Wait, the angle opposite to |a| would be the angle between b and a - b, which is 84°, the angle opposite to |b| would be the angle between a and a - b, which is θ, and the angle opposite to |a - b| would be the angle between a and b, which is 29°. Therefore, by the law of sines:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

But I don't know the magnitudes, so maybe I can relate them. Let me denote |a| = A, |b| = B, |a - b| = C. Then:

A / sin(84°) = B / sin(θ) = C / sin(29°)

But also, from the law of cosines:

C² = A² + B² - 2AB cos(29°)

But if I use the law of sines, I can express A and B in terms of C. Let me try that.

From the law of sines:

A = C * sin(84°) / sin(29°)

B = C * sin(θ) / sin(29°)

Therefore, substituting into the law of cosines:

C² = [C * sin(84°) / sin(29°)]² + [C * sin(θ) / sin(29°)]² - 2 * [C * sin(84°) / sin(29°)] * [C * sin(θ) / sin(29°)] * cos(29°)

Divide both sides by C²:

1 = [sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)] / sin²(29°)

Multiply both sides by sin²(29°):

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

This seems complicated, but maybe if θ is 67°, let's check if this equation holds.

First, compute sin²(29°):

sin(29°) ≈ 0.4848, so sin²(29°) ≈ 0.235

sin(84°) ≈ 0.9952, so sin²(84°) ≈ 0.9904

sin(67°) ≈ 0.9205, so sin²(67°) ≈ 0.8473

cos(29°) ≈ 0.8746

Now plug into the right-hand side:

sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°) ≈ 0.9904 + 0.8473 - 2 * 0.9952 * 0.9205 * 0.8746

Compute the last term:

First, 2 * 0.9952 * 0.9205 ≈ 2 * 0.915 ≈ 1.83

Then multiply by 0.8746: 1.83 * 0.8746 ≈ 1.60

So total RHS ≈ 0.9904 + 0.8473 - 1.60 ≈ 1.8377 - 1.60 ≈ 0.2377

Which is approximately equal to sin²(29°) ≈ 0.235. Close enough, considering rounding errors. Therefore, θ = 67° is indeed the correct answer. Therefore, the angle between a and a - b is 67 degrees.

But let me check if there's another way to approach this problem, just to confirm.

Alternatively, using vector algebra. Let me denote the angle between a and a - b as θ. Then, using the dot product:

cosθ = [a · (a - b)] / (|a| |a - b|)

Compute the numerator:

a · (a - b) = |a|² - a · b

We know that a · b = |a||b|cos(29°). So:

cosθ = [ |a|² - |a||b|cos(29°) ] / (|a| |a - b| )

Simplify numerator and denominator:

= [ |a| - |b|cos(29°) ] / |a - b|

But we also have from the previous information that the angle between b and a - b is 84°, so:

cos(84°) = [ b · (a - b) ] / ( |b| |a - b| )

Which we already calculated earlier as:

cos(84°) = [ a · b - |b|² ] / ( |b| |a - b| )

Which simplifies to:

cos(84°) = [ |a||b|cos(29°) - |b|² ] / ( |b| |a - b| )

= [ |a|cos(29°) - |b| ] / |a - b|

Therefore, we have two expressions:

From the angle between a and a - b:

cosθ = [ |a| - |b|cos(29°) ] / |a - b|  ...(1)

From the angle between b and a - b:

cos(84°) = [ |a|cos(29°) - |b| ] / |a - b|  ...(2)

Notice that equations (1) and (2) have the same denominator, |a - b|. Let me denote equation (1) as:

cosθ = [ |a| - |b|cos(29°) ] / |a - b|

and equation (2) as:

cos(84°) = [ |a|cos(29°) - |b| ] / |a - b|

Let me denote the numerator of equation (1) as N1 = |a| - |b|cos(29°)

and the numerator of equation (2) as N2 = |a|cos(29°) - |b|

So, cosθ = N1 / |a - b| and cos(84°) = N2 / |a - b|

Therefore, if I take the ratio of N1 to N2:

N1 / N2 = cosθ / cos(84°)

But N1 = |a| - |b|cos(29°)

N2 = |a|cos(29°) - |b|

Let me write this ratio:

[ |a| - |b|cos(29°) ] / [ |a|cos(29°) - |b| ] = cosθ / cos(84°)

Let me denote x = |a| / |b|. Let me set x = |a| / |b|, so |a| = x |b|. Then, substituting into the ratio:

[ x |b| - |b|cos(29°) ] / [ x |b| cos(29°) - |b| ] = [ x - cos(29°) ] / [ x cos(29°) - 1 ] = cosθ / cos(84°)

Therefore:

[ x - cos(29°) ] / [ x cos(29°) - 1 ] = cosθ / cos(84°)

Let me compute the left-hand side. Let me denote cos(29°) as c for simplicity. Then:

[ x - c ] / [ x c - 1 ] = cosθ / cos(84°)

Let me solve for x. Let me rearrange:

[ x - c ] / [ x c - 1 ] = cosθ / cos(84°)

Cross multiplying:

( x - c ) cos(84°) = ( x c - 1 ) cosθ

Expanding:

x cos(84°) - c cos(84°) = x c cosθ - cosθ

Bring all terms with x to one side:

x cos(84°) - x c cosθ = - cosθ + c cos(84°)

Factor x:

x [ cos(84°) - c cosθ ] = - cosθ + c cos(84°)

Therefore:

x = [ - cosθ + c cos(84°) ] / [ cos(84°) - c cosθ ]

But this seems complicated. However, if we assume θ = 67°, let's check if this equation holds. Let me compute c = cos(29°) ≈ 0.8746, cos(84°) ≈ 0.1045, cos(67°) ≈ 0.3907.

So substituting θ = 67°:

Numerator: -cos(67°) + c cos(84°) ≈ -0.3907 + 0.8746 * 0.1045 ≈ -0.3907 + 0.0914 ≈ -0.2993

Denominator: cos(84°) - c cos(67°) ≈ 0.1045 - 0.8746 * 0.3907 ≈ 0.1045 - 0.3417 ≈ -0.2372

Therefore, x ≈ (-0.2993)/(-0.2372) ≈ 1.262

So x ≈ 1.262. Let me check if this is consistent with the law of sines earlier. From the law of sines, we had:

A / sin(84°) = B / sin(θ) => x = A/B = sin(84°)/sin(θ) ≈ sin(84°)/sin(67°) ≈ 0.9952 / 0.9205 ≈ 1.081

Wait, but according to this calculation, x ≈ 1.262, which is different from 1.081. There's a discrepancy here. That suggests that maybe θ is not 67°? But earlier, using the triangle angle sum, we found θ = 67°, and the law of sines check with θ = 67° gave a close result. However, here, using this vector approach, there's inconsistency. What's wrong here?

Wait, maybe I made a mistake in the algebra. Let me check again.

We had:

From the ratio:

[ x - c ] / [ x c - 1 ] = cosθ / cos(84°)

If θ = 67°, then cosθ ≈ 0.3907, cos(84°) ≈ 0.1045, so the ratio is 0.3907 / 0.1045 ≈ 3.74

So [ x - c ] / [ x c - 1 ] ≈ 3.74

Let me compute [ x - c ] / [ x c - 1 ] with x ≈ 1.262:

Numerator: 1.262 - 0.8746 ≈ 0.3874

Denominator: 1.262 * 0.8746 - 1 ≈ 1.103 - 1 = 0.103

Therefore, ratio ≈ 0.3874 / 0.103 ≈ 3.76, which is close to 3.74. Considering rounding errors, this is consistent. Therefore, x ≈ 1.262 is acceptable. However, according to the law of sines, x should be sin(84°)/sin(θ) ≈ 0.9952 / sin(67°). Let me compute sin(67°): sin(67°) ≈ 0.9205, so 0.9952 / 0.9205 ≈ 1.081. But according to the previous calculation, x ≈ 1.262. These two values of x should be the same. Therefore, there's a contradiction here. Which means that my assumption that θ = 67° is wrong?

Wait, but earlier when I used the triangle angle sum, I thought θ = 67°, but according to this vector approach, there's inconsistency. Therefore, my initial assumption that the angles of the triangle are 29°, 84°, and θ is wrong. Therefore, my first approach was incorrect. So, I need to figure out where I went wrong.

Let me think again about the triangle. If I have vectors a, b, and a - b, then the triangle is formed by placing them head to tail. Wait, if I have vectors a and b, then the vector a - b is equivalent to a + (-b). So, if I place vector a and then vector -b, the resultant is a - b. Therefore, the triangle would have sides a, b, and a - b, but the angles between the vectors are not necessarily the internal angles of the triangle. Wait, maybe I need to think more carefully.

Let me consider the triangle formed by vectors a, b, and a - b. Let me imagine placing the tail of vector b at the tip of vector a. Then, the vector from the tail of a to the tip of b is a + b. But we need a - b. Alternatively, if I place the tail of vector a at the tip of vector b, then the vector from the tail of b to the tip of a is a - b. Therefore, the triangle would have sides |a|, |b|, and |a - b|, with the angle between a and b being 29°, the angle between b and a - b being 84°, and the angle between a and a - b being θ. However, in reality, the angle between a and a - b is not necessarily the internal angle of the triangle. Wait, maybe I need to use the law of cosines properly.

Alternatively, perhaps I need to use vector diagrams with coordinates. Let me assign coordinates to the vectors to calculate the angles.

Let me place vector a along the positive x-axis for simplicity. Let me denote vector a as (A, 0), where A is the magnitude of a. Then, vector b makes an angle of 29° with a, so its components would be (B cos(29°), B sin(29°)), where B is the magnitude of b.

Then, vector a - b would be (A - B cos(29°), -B sin(29°)).

Now, the angle between vector b and vector a - b is 84°, so we can use the dot product formula to find this angle.

First, compute the dot product of b and a - b:

b · (a - b) = |b| |a - b| cos(84°)

But also, from components:

b · (a - b) = [B cos(29°), B sin(29°)] · [A - B cos(29°), -B sin(29°)]

= B cos(29°)(A - B cos(29°)) + B sin(29°)(-B sin(29°))

= AB cos(29°) - B² cos²(29°) - B² sin²(29°)

= AB cos(29°) - B² [cos²(29°) + sin²(29°)]

= AB cos(29°) - B²

Which matches the earlier result. Therefore, we have:

AB cos(29°) - B² = |b| |a - b| cos(84°)

Which is:

B(A cos(29°) - B) = B |a - b| cos(84°)

Divide both sides by B:

A cos(29°) - B = |a - b| cos(84°)

Similarly, we can compute |a - b| from components:

|a - b|² = (A - B cos(29°))² + (-B sin(29°))²

= A² - 2AB cos(29°) + B² cos²(29°) + B² sin²(29°)

= A² - 2AB cos(29°) + B² [cos²(29°) + sin²(29°)]

= A² - 2AB cos(29°) + B²

Therefore, |a - b| = sqrt(A² + B² - 2AB cos(29°))

So, going back to the previous equation:

A cos(29°) - B = sqrt(A² + B² - 2AB cos(29°)) * cos(84°)

This is an equation relating A and B. Let me denote this as equation (3).

Now, we also need to find the angle between a and a - b. Let's compute that angle using the dot product.

Vector a is (A, 0), vector a - b is (A - B cos(29°), -B sin(29°))

Their dot product is:

a · (a - b) = A*(A - B cos(29°)) + 0*(-B sin(29°)) = A² - AB cos(29°)

The magnitude of a is A, and the magnitude of a - b is sqrt(A² + B² - 2AB cos(29°)) as before.

Therefore, the cosine of the angle θ between a and a - b is:

cosθ = [A² - AB cos(29°)] / [A * sqrt(A² + B² - 2AB cos(29°))]

Simplify numerator and denominator:

= [A - B cos(29°)] / sqrt(A² + B² - 2AB cos(29°))

Let me denote this as equation (4).

Now, from equation (3):

A cos(29°) - B = sqrt(A² + B² - 2AB cos(29°)) * cos(84°)

Let me denote sqrt(A² + B² - 2AB cos(29°)) as C, so:

A cos(29°) - B = C cos(84°)

From equation (4):

cosθ = [A - B cos(29°)] / C

Let me denote D = A - B cos(29°). Then, cosθ = D / C.

But from equation (3), we have A cos(29°) - B = C cos(84°). Let me see if I can relate D and A cos(29°) - B.

Let me write D = A - B cos(29°)

Let me also note that A cos(29°) - B = C cos(84°)

Let me try to find a relationship between D and A cos(29°) - B.

Let me consider D = A - B cos(29°)

Let me rearrange A cos(29°) - B:

A cos(29°) - B = cos(29°) * A - B

Compare with D = A - B cos(29°). These are different expressions. However, perhaps I can express A in terms of D and B cos(29°):

From D = A - B cos(29°), we have A = D + B cos(29°)

Substitute into A cos(29°) - B:

= [D + B cos(29°)] cos(29°) - B

= D cos(29°) + B cos²(29°) - B

= D cos(29°) + B [cos²(29°) - 1]

= D cos(29°) - B sin²(29°)

But from equation (3), this equals C cos(84°). Therefore:

D cos(29°) - B sin²(29°) = C cos(84°)

But D = [A - B cos(29°)] and C = sqrt(A² + B² - 2AB cos(29°))

This seems complicated. Maybe I can find a ratio between D and C?

From equation (4), cosθ = D / C, so D = C cosθ

Substitute into the previous equation:

C cosθ * cos(29°) - B sin²(29°) = C cos(84°)

Rearranged:

C cosθ cos(29°) - C cos(84°) = B sin²(29°)

Factor C:

C [cosθ cos(29°) - cos(84°)] = B sin²(29°)

But from equation (3), we have:

A cos(29°) - B = C cos(84°)

But A = D + B cos(29°) = C cosθ + B cos(29°)

Therefore:

[ C cosθ + B cos(29°) ] cos(29°) - B = C cos(84°)

Expand:

C cosθ cos(29°) + B cos²(29°) - B = C cos(84°)

Rearranged:

C cosθ cos(29°) - C cos(84°) = B - B cos²(29°)

Factor C on the left and B on the right:

C [cosθ cos(29°) - cos(84°)] = B [1 - cos²(29°)] = B sin²(29°)

Which matches the previous equation. Therefore, this doesn't give us new information. It seems like we need another equation to relate these variables.

Alternatively, let me consider dividing equation (3) by equation (4). Wait, equation (3) is:

A cos(29°) - B = C cos(84°)

Equation (4) is:

cosθ = D / C, where D = A - B cos(29°)

So, if I can express A cos(29°) - B in terms of D, maybe?

Let me note that:

Let me denote D = A - B cos(29°)

Let me also note that A cos(29°) - B = ?

Let me write A cos(29°) - B = ?

Let me express A from D:

A = D + B cos(29°)

Substitute into A cos(29°) - B:

= [D + B cos(29°)] cos(29°) - B

= D cos(29°) + B cos²(29°) - B

= D cos(29°) + B [cos²(29°) - 1]

= D cos(29°) - B sin²(29°)

So, equation (3) becomes:

D cos(29°) - B sin²(29°) = C cos(84°)

But from equation (4), D = C cosθ

Therefore:

C cosθ cos(29°) - B sin²(29°) = C cos(84°)

Rearranged:

C cosθ cos(29°) - C cos(84°) = B sin²(29°)

Factor C:

C [cosθ cos(29°) - cos(84°)] = B sin²(29°)

But from equation (4), D = C cosθ, and D = A - B cos(29°). Also, from equation (3), we have:

A cos(29°) - B = C cos(84°)

But A = D + B cos(29°) = C cosθ + B cos(29°)

Therefore:

[ C cosθ + B cos(29°) ] cos(29°) - B = C cos(84°)

Which expands to:

C cosθ cos(29°) + B cos²(29°) - B = C cos(84°)

Which rearranged is:

C cosθ cos(29°) - C cos(84°) = B - B cos²(29°)

Which is the same as before. So again, no new information.

This seems like a dead end. Maybe I need to assign specific values to A and B to solve for θ? Let me assume some value for B, say B = 1, then solve for A?

Let me try that. Let me set B = 1. Then, equation (3) becomes:

A cos(29°) - 1 = sqrt(A² + 1 - 2A cos(29°)) * cos(84°)

Let me denote cos(29°) ≈ 0.8746, cos(84°) ≈ 0.1045

So:

0.8746 A - 1 = sqrt(A² + 1 - 2 * 0.8746 A) * 0.1045

Let me denote the left side as L = 0.8746 A - 1

The right side is R = 0.1045 * sqrt(A² - 1.7492 A + 1)

Let me square both sides to eliminate the square root:

L² = R²

(0.8746 A - 1)² = (0.1045)² (A² - 1.7492 A + 1)

Compute left side:

≈ (0.7649 A² - 1.7492 A + 1)

Right side:

≈ 0.0109 * (A² - 1.7492 A + 1) ≈ 0.0109 A² - 0.01907 A + 0.0109

So:

0.7649 A² - 1.7492 A + 1 = 0.0109 A² - 0.01907 A + 0.0109

Bring all terms to left side:

0.7649 A² - 1.7492 A + 1 - 0.0109 A² + 0.01907 A - 0.0109 = 0

Combine like terms:

(0.7649 - 0.0109) A² + (-1.7492 + 0.01907) A + (1 - 0.0109) = 0

≈ 0.754 A² - 1.7301 A + 0.9891 = 0

Now, solve this quadratic equation for A:

A = [1.7301 ± sqrt( (1.7301)^2 - 4 * 0.754 * 0.9891 ) ] / (2 * 0.754)

Calculate discriminant:

(1.7301)^2 ≈ 2.993

4 * 0.754 * 0.9891 ≈ 4 * 0.746 ≈ 2.984

So discriminant ≈ 2.993 - 2.984 ≈ 0.009

Therefore, sqrt(0.009) ≈ 0.0949

Thus:

A ≈ [1.7301 ± 0.0949] / 1.508

Calculate both roots:

First root: (1.7301 + 0.0949)/1.508 ≈ 1.825 / 1.508 ≈ 1.210

Second root: (1.7301 - 0.0949)/1.508 ≈ 1.6352 / 1.508 ≈ 1.084

Now, check which one is valid. Remember that we squared the equation, so we might have extraneous solutions.

Let me check A ≈ 1.210:

Compute L = 0.8746 * 1.210 - 1 ≈ 1.058 - 1 = 0.058

Compute R = 0.1045 * sqrt( (1.210)^2 + 1 - 2 * 0.8746 * 1.210 )

First compute inside sqrt:

(1.464) + 1 - 2 * 0.8746 * 1.210 ≈ 2.464 - 2.103 ≈ 0.361

sqrt(0.361) ≈ 0.601

Then R ≈ 0.1045 * 0.601 ≈ 0.0628

Compare with L ≈ 0.058. Close, but not exact, likely due to rounding errors.

Now check A ≈ 1.084:

L = 0.8746 * 1.084 - 1 ≈ 0.947 - 1 = -0.053

R = 0.1045 * sqrt( (1.084)^2 + 1 - 2 * 0.8746 * 1.084 )

Compute inside sqrt:

≈ 1.175 + 1 - 2 * 0.8746 * 1.084 ≈ 2.175 - 1.903 ≈ 0.272

sqrt(0.272) ≈ 0.522

R ≈ 0.1045 * 0.522 ≈ 0.0545

Compare with L ≈ -0.053. Since L is negative and R is positive, this solution is extraneous. Therefore, the valid solution is A ≈ 1.210.

Now, with A ≈ 1.210 and B = 1, we can compute cosθ from equation (4):

cosθ = [A - B cos(29°)] / sqrt(A² + B² - 2AB cos(29°))

Compute numerator:

A - B cos(29°) ≈ 1.210 - 1 * 0.8746 ≈ 1.210 - 0.8746 ≈ 0.3354

Denominator:

sqrt(A² + B² - 2AB cos(29°)) ≈ sqrt(1.464 + 1 - 2 * 1.210 * 0.8746) ≈ sqrt(2.464 - 2.103) ≈ sqrt(0.361) ≈ 0.601

Therefore, cosθ ≈ 0.3354 / 0.601 ≈ 0.558

Then θ ≈ arccos(0.558) ≈ 56 degrees? Wait, arccos(0.558) is approximately 56 degrees? Let me check:

cos(56°) ≈ 0.5592, which is very close to 0.558. So θ ≈ 56 degrees. Wait, this contradicts the earlier result of 67 degrees. So which one is correct?

But earlier, when I assumed θ = 67°, the law of sines gave a consistent result with x ≈ 1.262, but with this coordinate method, I get θ ≈ 56°. There's a contradiction here. What's going on?

Wait, let me check my calculation of cosθ. With A ≈ 1.210, B = 1, angle between a and a - b:

cosθ = [A² - AB cos(29°)] / [A * |a - b|]

Wait, earlier I had:

cosθ = [A - B cos(29°)] / |a - b|

But |a - b| ≈ 0.601, and [A - B cos(29°)] ≈ 0.3354, so cosθ ≈ 0.3354 / 0.601 ≈ 0.558, which is about 56 degrees. But earlier, when I thought θ = 67°, I had a different value. So which is correct?

Let me check with θ ≈ 56°, what would be the angle between b and a - b?

From equation (3):

A cos(29°) - B = |a - b| cos(84°)

With A ≈ 1.210, B = 1, |a - b| ≈ 0.601:

Left side: 1.210 * 0.8746 - 1 ≈ 1.058 - 1 = 0.058

Right side: 0.601 * cos(84°) ≈ 0.601 * 0.1045 ≈ 0.0628

These are close, considering rounding errors. So this seems consistent. Therefore, θ ≈ 56°, not 67°. Therefore, my initial assumption that the triangle angles sum to 180° was wrong. So why was that?

Because the angles between the vectors are not the internal angles of the triangle. Wait, let me think again. If I have vectors a, b, and a - b, forming a triangle, then the angles between the vectors are not the internal angles of the triangle. For example, the angle between a and b is 29°, which is the angle between the vectors, but in the triangle, the angle opposite to side a - b would be the angle between a and b? Wait, no. Let me think.

If I have a triangle with sides |a|, |b|, and |a - b|, then the angle between sides |a| and |b| is the angle between vectors a and b, which is 29°. The angle between sides |b| and |a - b| is the angle between vector b and vector a - b, which is given as 84°. The angle between sides |a| and |a - b| is the angle between vector a and vector a - b, which is θ. Therefore, the three angles of the triangle are 29°, 84°, and θ, which should sum to 180°, so θ = 180° - 29° - 84° = 67°. But according to the coordinate calculation, θ ≈ 56°. There's a contradiction here. What is the problem?

Wait, maybe the angle between sides |b| and |a - b| in the triangle is not the same as the angle between vectors b and a - b? Wait, no. If the triangle is formed by placing the vectors head to tail, then the angle between sides |b| and |a - b| would be the angle between vector b and vector a - b. Wait, but in reality, when you form a triangle with vectors a, b, and a - b, the angle between b and a - b is not necessarily the internal angle of the triangle. Wait, let me clarify.

Let me consider the triangle with vertices at points O, A, and B, where vector OA = a, vector OB = b, and vector AB = a - b. Then, the angle at point B between vectors BA and BC (wait, no). Wait, the triangle is OAB, with OA = a, OB = b, and AB = a - b. The angle between vectors b and a - b would be the angle at point A between vectors AB and AO? Wait, this is getting confusing.

Alternatively, let me think of the triangle with sides OA = a, OB = b, and AB = a - b. The angle between vectors OA and OB is 29°, which is the angle at point O. The angle between vectors OB and AB is the angle at point B between OB and AB. But vector AB is a - b, which is from A to B. Wait, vector AB is b - a, not a - b. Wait, confusion arises here.

Wait, vector AB is from point A to point B, which is OB - OA = b - a. Therefore, vector AB = b - a, so vector a - b = -AB. Therefore, the angle between vector b and vector a - b is the angle between vector b and -AB. Which is supplementary to the angle between vector b and AB. Therefore, if the angle between b and AB is φ, then the angle between b and a - b is 180° - φ.

Therefore, the angle between b and a - b is not the same as the internal angle of the triangle. Therefore, my initial assumption that the angles of the triangle are 29°, 84°, and θ is incorrect. Therefore, the triangle angle sum approach was wrong. Hence, the correct answer is approximately 56°, which is about 56 degrees. But let me check with more precise calculations.

Earlier, with A ≈ 1.210, B = 1, we found θ ≈ 56°. Let me calculate it more precisely.

cosθ ≈ 0.3354 / 0.601 ≈ 0.558. Let me compute arccos(0.558):

Using calculator, arccos(0.558) ≈ 56.1 degrees. So approximately 56 degrees. But the problem says to find the exact value. How?

Wait, maybe there's an exact relationship here. Let me think again.

From the coordinate method, we found that θ ≈ 56°, but how to find the exact value?

Alternatively, maybe there's a relationship between the angles. Let me consider the triangle formed by vectors a, b, and a - b. Wait, but earlier confusion about the angles. Let me try to use the law of sines properly.

In the triangle with sides |a|, |b|, |a - b|, the angles opposite these sides are:

- Angle opposite |a|: let's call it α, which is the angle between b and a - b.

- Angle opposite |b|: let's call it β, which is the angle between a and a - b (which is θ).

- Angle opposite |a - b|: let's call it γ, which is the angle between a and b, which is 29°.

Therefore, by the law of sines:

|a| / sin(α) = |b| / sin(β) = |a - b| / sin(γ)

We know γ = 29°, α is the angle between b and a - b, which is given as 84°, and β is θ, which we need to find.

Therefore:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

From this, we can write:

|a| / |b| = sin(84°) / sin(θ)

and

|a - b| / |b| = sin(29°) / sin(θ)

Let me denote |a| = k |b|, so k = |a| / |b| = sin(84°) / sin(θ)

Also, |a - b| = |b| * sin(29°) / sin(θ)

But from the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Substitute |a| = k |b| and |a - b| = |b| * sin(29°) / sin(θ):

[ |b|² * sin²(29°) / sin²(θ) ] = k² |b|² + |b|² - 2 * k |b|² * cos(29°)

Divide both sides by |b|²:

sin²(29°) / sin²(θ) = k² + 1 - 2k cos(29°)

But k = sin(84°) / sin(θ), so substitute:

sin²(29°) / sin²(θ) = [sin²(84°) / sin²(θ)] + 1 - 2 [sin(84°) / sin(θ)] cos(29°)

Multiply both sides by sin²(θ):

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

This is the same equation as before. Let me rearrange:

sin²(θ) - 2 sin(84°) cos(29°) sin(θ) + sin²(84°) - sin²(29°) = 0

This is a quadratic equation in sin(θ):

Let me denote x = sin(θ). Then:

x² - 2 sin(84°) cos(29°) x + [sin²(84°) - sin²(29°)] = 0

Let me compute the coefficients:

First, compute 2 sin(84°) cos(29°). Using the identity 2 sin A cos B = sin(A + B) + sin(A - B):

2 sin(84°) cos(29°) = sin(84° + 29°) + sin(84° - 29°) = sin(113°) + sin(55°)

Similarly, sin²(84°) - sin²(29°) = [sin(84°) - sin(29°)][sin(84°) + sin(29°)]

But maybe it's easier to compute numerically.

Compute 2 sin(84°) cos(29°):

sin(84°) ≈ 0.9952, cos(29°) ≈ 0.8746

So 2 * 0.9952 * 0.8746 ≈ 2 * 0.870 ≈ 1.740

Next, sin²(84°) - sin²(29°):

sin²(84°) ≈ 0.9904, sin²(29°) ≈ 0.235, so difference ≈ 0.9904 - 0.235 ≈ 0.7554

Therefore, the quadratic equation is:

x² - 1.740 x + 0.7554 = 0

Solving for x:

x = [1.740 ± sqrt(1.740² - 4 * 1 * 0.7554)] / 2

Calculate discriminant:

1.740² ≈ 3.0276

4 * 0.7554 ≈ 3.0216

Discriminant ≈ 3.0276 - 3.0216 ≈ 0.006

sqrt(0.006) ≈ 0.0775

Thus:

x ≈ [1.740 ± 0.0775] / 2

First solution: (1.740 + 0.0775)/2 ≈ 1.8175 / 2 ≈ 0.9087

Second solution: (1.740 - 0.0775)/2 ≈ 1.6625 / 2 ≈ 0.8312

So sin(θ) ≈ 0.9087 or 0.8312

Therefore, θ ≈ arcsin(0.9087) ≈ 65° or arcsin(0.8312) ≈ 56.2°

Wait, but earlier with the coordinate method, we found θ ≈ 56°, which corresponds to sin(θ) ≈ 0.8312. The other solution, sin(θ) ≈ 0.9087, would correspond to θ ≈ 65° or 180° - 65° = 115°. But since θ is an angle between two vectors, it must be between 0° and 180°, so both are possible. How to determine which one is correct?

Let me check with the previous coordinate calculation where θ ≈ 56°, which gave sin(θ) ≈ 0.829, close to 0.8312. The other solution, θ ≈ 65°, would have sin(θ) ≈ 0.9063, close to 0.9087. So both are possible. How to determine which one is correct?

Let me recall that in the coordinate method, we found that with A ≈ 1.210, B = 1, the angle θ ≈ 56°, which corresponds to the smaller solution. But why does the quadratic have two solutions?

This is because the equation sin²(θ) - 2 sin(84°) cos(29°) sin(θ) + [sin²(84°) - sin²(29°)] = 0 is derived from the law of sines and law of cosines, and it can have two solutions depending on the configuration. However, we need to determine which one is valid based on the given angles.

Let me think about the triangle. If θ ≈ 56°, then the angles of the triangle would be γ = 29°, α = 84°, and β = 56°, which sum to 29 + 84 + 56 = 169°, which is not 180°. Wait, this contradicts the triangle angle sum. Wait, but earlier I thought that α is the angle opposite |a|, which is the angle between b and a - b, which is 84°, and β is the angle opposite |b|, which is the angle between a and a - b, which is θ. And γ is the angle opposite |a - b|, which is the angle between a and b, which is 29°. So the sum should be 84° + θ + 29° = 180°, so θ = 180° - 84° - 29° = 67°. But according to the quadratic equation, θ could be 56° or 65°, which don't add up to 180° with 84° and 29°. This suggests a contradiction.

Wait, this indicates that my earlier assumption about which angle corresponds to which side is incorrect. Let me re-examine the law of sines.

In the triangle with sides |a|, |b|, |a - b|, the angles opposite these sides are:

- The angle opposite |a| is the angle between b and a - b, which is given as 84°.

- The angle opposite |b| is the angle between a and a - b, which is θ.

- The angle opposite |a - b| is the angle between a and b, which is 29°.

Therefore, the sum of angles should be 84° + θ + 29° = 180°, so θ = 67°. But according to the quadratic equation, θ could be 56° or 65°, which suggests inconsistency. What is the problem here?

Wait, the law of sines says that |a| / sin(α) = |b| / sin(β) = |a - b| / sin(γ), where α = 84°, β = θ, γ = 29°. So, if we use this, then:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

But from the law of cosines, we also have:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

So combining these, we derived the quadratic equation, which gave two solutions for sin(θ). But if the triangle angle sum requires θ = 67°, then why does the quadratic give different results?

This suggests that there is an inconsistency, which likely arises from an error in assuming that the angle opposite |a| is 84°. Let me verify this.

If the angle opposite |a| is the angle between b and a - b, which is 84°, then the law of sines should hold. But if we calculate with θ = 67°, let's see if it satisfies the quadratic equation.

Let me compute sin(θ) = sin(67°) ≈ 0.9205

Plug into the quadratic equation:

x² - 1.740 x + 0.7554 = 0

With x = 0.9205:

(0.9205)^2 - 1.740 * 0.9205 + 0.7554 ≈ 0.8473 - 1.601 + 0.7554 ≈ 0.8473 + 0.7554 - 1.601 ≈ 1.6027 - 1.601 ≈ 0.0017 ≈ 0. This is very close to zero, considering rounding errors. Therefore, θ = 67° is a solution!

Wait, earlier when I solved the quadratic equation, I got x ≈ 0.9087 and 0.8312, but if θ = 67°, sin(θ) ≈ 0.9205, which is close to 0.9087? Wait, no, 0.9205 is not close to 0.9087. Wait, what's going on?

Wait, let me recalculate the quadratic equation with more precision.

Earlier, I approximated the coefficients as:

2 sin(84°) cos(29°) ≈ 1.740

sin²(84°) - sin²(29°) ≈ 0.7554

But let me calculate these more accurately.

First, calculate 2 sin(84°) cos(29°):

sin(84°) = sin(90° - 6°) = cos(6°) ≈ 0.994521895

cos(29°) ≈ 0.874619707

So 2 * 0.994521895 * 0.874619707 ≈ 2 * 0.8693 ≈ 1.7386

Next, sin²(84°) - sin²(29°):

sin(84°) ≈ 0.994521895, so sin²(84°) ≈ 0.98907

sin(29°) ≈ 0.48480962, so sin²(29°) ≈ 0.2350

Thus, sin²(84°) - sin²(29°) ≈ 0.98907 - 0.2350 ≈ 0.75407

So the quadratic equation is:

x² - 1.7386x + 0.75407 = 0

Now, let's solve this with more precision.

Discriminant D = (1.7386)^2 - 4 * 1 * 0.75407

Calculate 1.7386²: 1.7386 * 1.7386 ≈ 3.022

4 * 0.75407 ≈ 3.01628

Thus, D ≈ 3.022 - 3.01628 ≈ 0.00572

sqrt(D) ≈ 0.0756

Solutions:

x = [1.7386 ± 0.0756]/2

First solution: (1.7386 + 0.0756)/2 ≈ 1.8142 / 2 ≈ 0.9071

Second solution: (1.7386 - 0.0756)/2 ≈ 1.663 / 2 ≈ 0.8315

Now, if θ = 67°, sin(θ) ≈ 0.9205, which is not equal to 0.9071 or 0.8315. Wait, but earlier when I plugged θ = 67° into the equation, it gave approximately zero. Let me check again.

Let me compute x = sin(67°) ≈ 0.9205

Plug into x² - 1.7386x + 0.75407:

(0.9205)^2 - 1.7386 * 0.9205 + 0.75407 ≈ 0.8473 - 1.601 + 0.75407 ≈ 0.8473 + 0.75407 - 1.601 ≈ 1.60137 - 1.601 ≈ 0.00037, which is approximately zero. So θ = 67° is a solution! But why does the quadratic equation give x ≈ 0.9071 and 0.8315? Because sin(67°) ≈ 0.9205 is not one of the solutions? Wait, this suggests that there is a calculation error.

Wait, let me check the quadratic equation again. The equation was derived from:

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

But if θ = 67°, let's compute both sides:

Left side: sin²(29°) ≈ 0.2350

Right side: sin²(84°) + sin²(67°) - 2 sin(84°) sin(67°) cos(29°)

Compute each term:

sin²(84°) ≈ 0.98907

sin²(67°) ≈ 0.8473

2 sin(84°) sin(67°) cos(29°) ≈ 2 * 0.9945 * 0.9205 * 0.8746 ≈ 2 * 0.9945 * 0.9205 * 0.8746

First compute 0.9945 * 0.9205 ≈ 0.915

Then 0.915 * 0.8746 ≈ 0.799

Multiply by 2: ≈ 1.598

So right side ≈ 0.98907 + 0.8473 - 1.598 ≈ 1.83637 - 1.598 ≈ 0.23837

Left side is 0.2350, which is close to 0.23837, considering rounding errors. Therefore, θ = 67° is a solution. But why does the quadratic equation give x ≈ 0.9071 and 0.8315? Because when I solved the quadratic equation, I used approximate values for the coefficients, leading to approximate solutions. The exact solution is θ = 67°, and the other solution is an extraneous one due to the quadratic nature.

Wait, but how can θ = 67° satisfy the quadratic equation? Let me check with x = sin(67°) ≈ 0.9205:

Plug into x² - 1.7386x + 0.75407 ≈ 0.8473 - 1.7386 * 0.9205 + 0.75407 ≈ 0.8473 - 1.601 + 0.75407 ≈ 0.8473 + 0.75407 - 1.601 ≈ 1.60137 - 1.601 ≈ 0.00037, which is approximately zero. So θ = 67° is a solution. But why does the quadratic equation give x ≈ 0.9071 and 0.8315? Because when I solved the quadratic equation, I used approximate values for the coefficients, leading to approximate solutions. The exact solution is θ = 67°, and the other solution is due to the quadratic having two roots, but one of them is extraneous.

Wait, but how can θ = 67° be a solution if the triangle angle sum suggests θ = 67°, and the quadratic equation also gives it as a solution? Earlier, when I did the coordinate method, I got θ ≈ 56°, but that must have been due to rounding errors in the calculation. Let me check with θ = 67°, what would be the value of A and B?

Let me assume θ = 67°, and use the law of sines:

|a| / sin(84°) = |b| / sin(67°) = |a - b| / sin(29°)

Let me set |b| = 1, then |a| = sin(84°)/sin(67°) ≈ 0.9945 / 0.9205 ≈ 1.080

|a - b| = sin(29°)/sin(67°) ≈ 0.4848 / 0.9205 ≈ 0.5267

Now, let's check the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Left side: (0.5267)^2 ≈ 0.2774

Right side: (1.080)^2 + 1^2 - 2 * 1.080 * 1 * cos(29°) ≈ 1.166 + 1 - 2 * 1.080 * 0.8746 ≈ 2.166 - 1.883 ≈ 0.283

These are close, considering rounding errors. Therefore, θ = 67° is indeed a valid solution. Earlier, when I did the coordinate method, I got θ ≈ 56°, but that must have been due to an error in calculation. Let me check that again.

Previously, I set B = 1, and found A ≈ 1.210, but with θ = 67°, |a| = 1.080, which is different. Let me redo the coordinate method with θ = 67°.

Let me set B = 1, and |a| = 1.080. Then, vector a is (1.080, 0), vector b is (cos(29°), sin(29°)) ≈ (0.8746, 0.4848). Then, vector a - b is (1.080 - 0.8746, 0 - 0.4848) ≈ (0.2054, -0.4848)

Now, compute the angle between b and a - b. Let's compute the dot product:

b · (a - b) = (0.8746)(0.2054) + (0.4848)(-0.4848) ≈ 0.1793 - 0.235 ≈ -0.0557

The magnitude of b is 1, and the magnitude of a - b is sqrt(0.2054² + (-0.4848)^2) ≈ sqrt(0.0422 + 0.235) ≈ sqrt(0.2772) ≈ 0.5267

Therefore, cos(angle) = -0.0557 / (1 * 0.5267) ≈ -0.1057

Which corresponds to an angle of arccos(-0.1057) ≈ 96°, but the problem states that the angle between b and a - b is 84°. This is a contradiction! What's wrong here?

Wait, this suggests that θ = 67° leads to the angle between b and a - b being 96°, not 84°, which contradicts the given information. Therefore, θ = 67° is not the correct solution. This is very confusing.

Wait, but earlier, when I plugged θ = 67° into the equation derived from the law of sines and law of cosines, it gave a near-zero result. What is the issue here?

Let me check the calculation of the dot product. Vector b is (0.8746, 0.4848), vector a - b is (0.2054, -0.4848). Their dot product is:

0.8746 * 0.2054 + 0.4848 * (-0.4848) ≈ 0.1793 - 0.235 ≈ -0.0557

But according to the earlier formula, b · (a - b) = AB cos(29°) - B² = 1.080 * 1 * cos(29°) - 1² ≈ 1.080 * 0.8746 - 1 ≈ 0.9446 - 1 ≈ -0.0554, which matches the dot product calculation. So this is correct.

Now, the angle between b and a - b is arccos(-0.0557 / (|b| |a - b|)) = arccos(-0.0557 / (1 * 0.5267)) ≈ arccos(-0.1057) ≈ 96°, but the problem states it should be 84°. This is a contradiction. What is the problem here?

Wait, this suggests that θ = 67° is not the correct solution, but it satisfies the equation derived from the law of sines and law of cosines. How is this possible?

Wait, let me check the angle between b and a - b. If the dot product is negative, the angle is obtuse. The problem states it is 84°, which is acute. This suggests that there is a mistake in the assumption that the angle between b and a - b is 84°, but in reality, it is 180° - 84° = 96°, which matches the calculation. This implies that the given angle between b and a - b is actually the angle between b and - (a - b), which would be 84°, but the actual angle between b and a - b is 96°. This suggests that there is a sign error in the dot product.

Wait, the angle between two vectors is always between 0° and 180°, and the dot product can be positive or negative. If the dot product is negative, the angle is obtuse. But the problem states that the angle between b and a - b is 84°, which is acute. This suggests that there is a mistake in the calculation.

Wait, let me re-examine the problem statement: "the angle between b and a - b is 84°." If the dot product is negative, the angle is obtuse, which contradicts the given 84°. Therefore, this suggests that θ = 67° is not the correct solution, and the other solution, θ ≈ 56°, is the correct one.

Let me check with θ ≈ 56°. Let me take the solution from the quadratic equation where sin(θ) ≈ 0.8315, so θ ≈ 56.2°. Let me set θ = 56.2°, and use the law of sines to find |a| and |a - b|.

Let me set |b| = 1, then |a| = sin(84°)/sin(θ) ≈ 0.9945 / sin(56.2°). Let me calculate sin(56.2°). 56.2° is approximately 56°, sin(56°) ≈ 0.8290, so let's say sin(θ) ≈ 0.8290. Then |a| ≈ 0.9945 / 0.8290 ≈ 1.200.

Then |a - b| = sin(29°)/sin(θ) ≈ 0.4848 / 0.8290 ≈ 0.5847

Now, let's check the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Left side: (0.5847)^2 ≈ 0.342

Right side: (1.200)^2 + 1^2 - 2 * 1.200 * 1 * 0.8746 ≈ 1.44 + 1 - 2.099 ≈ 2.44 - 2.099 ≈ 0.341, which matches.

Now, let's compute the angle between b and a - b. Using the dot product:

b · (a - b) = AB cos(29°) - B² = 1.200 * 1 * 0.8746 - 1 ≈ 1.0495 - 1 = 0.0495

The magnitude of a - b is 0.5847, so cos(angle) = 0.0495 / (1 * 0.5847) ≈ 0.0847

Therefore, the angle is arccos(0.0847) ≈ 85°, which is close to the given 84°, considering rounding errors. This suggests that θ ≈ 56° is the correct solution, and the earlier assumption that θ = 67° was incorrect due to a miscalculation in the angle between b and a - b.

This indicates that the correct angle is approximately 56°, and the earlier triangle angle sum approach was incorrect because the angle between b and a - b is not the internal angle of the triangle, but rather the angle between the vector b and the vector a - b, which can be obtuse or acute depending on the configuration.

Therefore, the correct answer is approximately 56°, but we need to find the exact value.

Let me return to the quadratic equation:

x² - 1.7386x + 0.75407 = 0, where x = sin(θ)

We found two solutions: x ≈ 0.9071 and x ≈ 0.8315. We determined that x ≈ 0.8315 corresponds to θ ≈ 56.2°, which is the correct solution. Let me find the exact value.

Let me recall that the quadratic equation was derived from:

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

Let me rearrange this equation:

2 sin(84°) sin(θ) cos(29°) = sin²(84°) + sin²(θ) - sin²(29°)

Let me use the identity sin²(A) - sin²(B) = sin(A + B) sin(A - B)

So, sin²(84°) - sin²(29°) = sin(84° + 29°) sin(84° - 29°) = sin(113°) sin(55°)

Therefore, the equation becomes:

2 sin(84°) sin(θ) cos(29°) = sin(113°) sin(55°) + sin²(θ)

Let me note that sin(113°) = sin(180° - 67°) = sin(67°), and sin(55°) is as is.

So:

2 sin(84°) sin(θ) cos(29°) = sin(67°) sin(55°) + sin²(θ)

Let me rearrange:

sin²(θ) - 2 sin(84°) cos(29°) sin(θ) + sin(67°) sin(55°) = 0

This is the same as before. Let me try to find an exact solution.

Let me use the values:

sin(67°) ≈ 0.9205, sin(55°) ≈ 0.8192, sin(84°) ≈ 0.9952, cos(29°) ≈ 0.8746

So:

sin²(θ) - 2 * 0.9952 * 0.8746 * sin(θ) + 0.9205 * 0.8192 ≈ 0

Calculate the coefficients:

2 * 0.9952 * 0.8746 ≈ 1.7386

0.9205 * 0.8192 ≈ 0.7540

So the equation is:

sin²(θ) - 1.7386 sin(θ) + 0.7540 = 0

Which is the same as before. Let me try to solve this equation using exact trigonometric identities.

Let me recall that 84° = 67° + 17°, but not sure if that helps. Alternatively, let me consider that 29° + 84° = 113°, and 84° - 29° = 55°, which we already used.

Alternatively, let me assume that θ = 56°, and check if it satisfies the equation.

sin(56°) ≈ 0.8290

sin²(56°) ≈ 0.687

1.7386 * 0.8290 ≈ 1.437

0.7540

So 0.687 - 1.437 + 0.7540 ≈ 0.687 + 0.7540 - 1.437 ≈ 1.441 - 1.437 ≈ 0.004, which is close to zero. This suggests θ ≈ 56° is the solution.

But how to find the exact value? Let me think about the given angles: 29° and 84°. Note that 84° = 90° - 6°, and 29° is given. Maybe there is a relationship here.

Alternatively, let me consider the triangle formed by the vectors. If the angle between a and a - b is θ, and the angle between b and a - b is 84°, and the angle between a and b is 29°, then using the law of sines in the triangle, we have:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

Let me denote |a| = k |b|, so k = |a| / |b| = sin(84°) / sin(θ)

Also, |a - b| = |b| * sin(29°) / sin(θ)

From the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Substitute |a| = k |b| and |a - b| = |b| * sin(29°) / sin(θ):

[ sin²(29°) / sin²(θ) ] |b|² = k² |b|² + |b|² - 2k |b|² cos(29°)

Divide by |b|²:

sin²(29°) / sin²(θ) = k² + 1 - 2k cos(29°)

But k = sin(84°) / sin(θ), so:

sin²(29°) / sin²(θ) = [sin²(84°) / sin²(θ)] + 1 - 2 [sin(84°) / sin(θ)] cos(29°)

Multiply both sides by sin²(θ):

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

This is the same equation as before. Let me try to solve this equation for θ.

Let me use the identity for sin(A) - sin(B) or other identities. Let me rearrange:

2 sin(84°) sin(θ) cos(29°) = sin²(84°) + sin²(θ) - sin²(29°)

Let me use the identity sin²(84°) - sin²(29°) = sin(84° + 29°) sin(84° - 29°) = sin(113°) sin(55°)

So:

2 sin(84°) sin(θ) cos(29°) = sin(113°) sin(55°) + sin²(θ)

Let me note that sin(113°) = sin(67°), so:

2 sin(84°) sin(θ) cos(29°) = sin(67°) sin(55°) + sin²(θ)

Let me try to express this in terms of θ. Let me recall that 84° = 67° + 17°, and 55° = 67° - 12°, but not sure.

Alternatively, let me use the identity for 2 sin A cos B = sin(A + B) + sin(A - B). Let me apply this to 2 sin(84°) cos(29°):

2 sin(84°) cos(29°) = sin(84° + 29°) + sin(84° - 29°) = sin(113°) + sin(55°) = sin(67°) + sin(55°)

Therefore, the equation becomes:

[sin(67°) + sin(55°)] sin(θ) = sin(67°) sin(55°) + sin²(θ)

Let me expand the left side:

sin(67°) sin(θ) + sin(55°) sin(θ) = sin(67°) sin(55°) + sin²(θ)

Rearrange:

sin(67°) sin(θ) + sin(55°) sin(θ) - sin(67°) sin(55°) - sin²(θ) = 0

Factor terms:

sin(67°) [sin(θ) - sin(55°)] + sin(55°) sin(θ) - sin²(θ) = 0

Wait, this seems complex. Let me group terms differently:

[sin(67°) sin(θ) - sin(67°) sin(55°)] + [sin(55°) sin(θ) - sin²(θ)] = 0

Factor sin(67°) from the first group and sin(θ) from the second group:

sin(67°)[sin(θ) - sin(55°)] + sin(θ)[sin(55°) - sin(θ)] = 0

Notice that [sin(θ) - sin(55°)] = -[sin(55°) - sin(θ)], so:

sin(67°)[sin(θ) - sin(55°)] - sin(θ)[sin(θ) - sin(55°)] = 0

Factor out [sin(θ) - sin(55°)]:

[sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0

Therefore, either:

1. sin(θ) - sin(55°) = 0 => sin(θ) = sin(55°) => θ = 55° or 125°

2. sin(67°) - sin(θ) = 0 => sin(θ) = sin(67°) => θ = 67° or 113°

Now, let's check these solutions.

First, θ = 55°: let's check if it satisfies the original equation.

From the equation: sin²(29°) = sin²(84°) + sin²(55°) - 2 sin(84°) sin(55°) cos(29°)

Compute left side: sin²(29°) ≈ 0.2350

Right side: sin²(84°) + sin²(55°) - 2 sin(84°) sin(55°) cos(29°)

Compute each term:

sin²(84°) ≈ 0.98907

sin²(55°) ≈ (0.8192)^2 ≈ 0.6710

2 sin(84°) sin(55°) cos(29°) ≈ 2 * 0.9945 * 0.8192 * 0.8746 ≈ 2 * 0.9945 * 0.716 ≈ 2 * 0.711 ≈ 1.422

So right side ≈ 0.98907 + 0.6710 - 1.422 ≈ 1.660 - 1.422 ≈ 0.238, which is close to 0.2350, considering rounding errors. So θ = 55° is a solution.

Second, θ = 67°: as before, it also satisfies the equation.

Third, θ = 125°: let's check.

sin(125°) = sin(55°) ≈ 0.8192, so it would be the same as θ = 55°, but since sin(θ) is the same, it would give the same result. Similarly, θ = 113° would be sin(113°) = sin(67°), which would be the same as θ = 67°.

Now, we need to determine which of these solutions is valid based on the given information.

If θ = 55°, let's check the angle between b and a - b. Let me use the law of sines with θ = 55°.

Let |b| = 1, then |a| = sin(84°)/sin(55°) ≈ 0.9945 / 0.8192 ≈ 1.214

|a - b| = sin(29°)/sin(55°) ≈ 0.4848 / 0.8192 ≈ 0.592

Now, compute the angle between b and a - b using the dot product:

b · (a - b) = |a||b|cos(29°) - |b|² = 1.214 * 1 * 0.8746 - 1 ≈ 1.061 - 1 = 0.061

The magnitude of a - b is 0.592, so cos(angle) = 0.061 / (1 * 0.592) ≈ 0.103, which corresponds to an angle of arccos(0.103) ≈ 84°, which matches the given information! This is the correct solution.

Similarly, if θ = 67°, as before, the angle between b and a - b was 96°, which does not match the given 84°. Therefore, the correct solution is θ = 55°.

Wait, but earlier when I solved the quadratic equation, I got θ ≈ 56°, which is close to 55°. This suggests that the exact value is 55°, and the small discrepancy is due to rounding.

Let me verify with θ = 55°:

From the law of sines, the angles of the triangle would be:

- Angle opposite |a|: 84° (between b and a - b)

- Angle opposite |b|: 55° (between a and a - b)

- Angle opposite |a - b|: 29° (between a and b)

Sum: 84 + 55 + 29 = 168°, which is not 180°. Wait, this is a contradiction! What is going on?

Wait, no! The angle opposite |a - b| is the angle between a and b, which is 29°, and the other two angles are 84° and 55°, which sum to 139°, not 180°. This suggests a mistake in the assumption about the angles in the triangle.

Wait, this indicates that the law of sines approach is not directly giving the angles of the triangle, but rather the angles between the vectors. This is the confusion I had earlier. The angle between a and b is 29°, which is the angle between vectors a and b, but in the triangle, the angle opposite |a - b| is the angle between a and b, which is 29°. The angle opposite |a| is the angle between b and a - b, which is 84°, and the angle opposite |b| is the angle between a and a - b, which is θ = 55°. Therefore, the sum of angles in the triangle is 29° + 84° + 55° = 168°, which is not 180°. This is impossible, so there must be a mistake.

Wait, this suggests that the law of sines is not being applied correctly. Let me recall that in the triangle, the angles are the angles between the sides, not the angles between the vectors. The angle between vectors a and b is 29°, which is the angle between the sides |a| and |b| in the triangle. The angle between vectors b and a - b is 84°, which is the angle between the sides |b| and |a - b| in the triangle. The angle between vectors a and a - b is θ, which is the angle between the sides |a| and |a - b| in the triangle. Therefore, the sum of these three angles should be 180°, so 29° + 84° + θ = 180°, giving θ = 67°. But this contradicts the previous result of θ = 55°. What is the resolution?

The resolution is that the angle between vectors b and a - b is not the same as the angle between sides |b| and |a - b| in the triangle. This is because the vector a - b is not the side of the triangle, but the vector from the tip of b to the tip of a. Therefore, the angle between b and a - b is not the internal angle of the triangle, but rather an external angle or something else. This is why the law of sines approach gave a contradiction.

Therefore, the correct approach is the one using the quadratic equation, which gave θ ≈ 55°, and this satisfies the given condition that the angle between b and a - b is 84°. The earlier triangle angle sum approach was incorrect because it assumed the angles between vectors are the internal angles of the triangle, which is not the case.

Therefore, the correct angle is θ = 55°, but let me check with the exact calculation.

From the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0, we have two solutions: θ = 55° and θ = 67°. We determined that θ = 55° is the correct solution, as it satisfies the given angle between b and a - b being 84°. However, 55° is not one of the standard angles, but let me check if it's 55° or 56°.

Wait, when I calculated with θ = 55°, the angle between b and a - b was 84°, which matches the problem statement. Therefore, the correct answer is 55°. But let me check with θ = 55°, what is the exact value?

Wait, let me calculate the exact value using the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0. The solutions are θ = 55° and θ = 67°. But why does θ = 55° work?

Let me recall that when θ = 55°, the angle between b and a - b is 84°, which is given. When θ = 67°, the angle between b and a - b is 96°, which is not given. Therefore, θ = 55° is the correct solution.

But why does the triangle angle sum give 29 + 84 + 55 = 168°? Because the angle between vectors a and b is 29°, which is the angle between sides |a| and |b| in the triangle, the angle between vectors b and a - b is 84°, which is not the internal angle of the triangle, but the angle between vector b and vector a - b, which is different from the internal angle of the triangle. Therefore, the triangle angle sum is not directly applicable here.

Therefore, the correct answer is θ = 55°. But let me check with the exact calculation.

Wait, when I used θ = 55°, the angle between b and a - b was 84°, which matches the problem statement. Therefore, the correct answer is 55°. But let me check if 55° is the exact value or if it's a rounded value.

Let me recall that in the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0, the solutions are θ = 55° and θ = 67°. Since θ = 55° satisfies the given condition, it is the correct solution. Therefore, the angle between a and a - b is 55°.

But let me check with the initial problem statement. The angle between a and b is 29°, and the angle between b and a - b is 84°. If the angle between a and a - b is 55°, then the three angles in the triangle would be 29°, 84°, and 55°, but this sums to 168°, which is not 180°. This suggests a contradiction, but it is resolved by understanding that the angle between b and a - b is not the internal angle of the triangle.

Therefore, the correct answer is 55°. However, let me check with the exact calculation using the law of sines and the given angles.

Let me use θ = 55°, and calculate the angle between a and a - b as 55°, and verify the given angle between b and a - b is 84°.

As done earlier, with |b| = 1, |a| ≈ 1.214, |a - b| ≈ 0.592, the dot product between b and a - b is 0.061, and the angle is arccos(0.061 / (1 * 0.592)) ≈ arccos(0.103) ≈ 84°, which matches. Therefore, θ = 55° is correct.

But why does the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0 give θ = 55° as a solution? Because it is derived from the original equation, and it satisfies the given conditions.

Therefore, the correct answer is 55°. But let me check if 55° is the exact value or if it's a rounded value. Let me calculate the exact value.

Let me recall that in the equation, we had:

[sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0

This suggests that θ = 55° or θ = 67°. Since θ = 55° is the correct solution, the answer is 55°. However, let me check if 55° is the exact value or if it's an approximation.

Let me calculate the exact value using the given angles. Let me recall that 29° + 84° = 113°, and 55° + 67° = 122°, but not sure.

Alternatively, let me consider that 55° + 29° = 84°, which is the given angle between b and a - b. This suggests a relationship: the angle between a and a - b (55°) plus the angle between a and b (29°) equals the angle between b and a - b (84°). This is a possible relationship, but it's not a standard identity.

However, this suggests that the angle between a and a - b is 55°, which is 84° - 29°. Let me verify this:

If θ = 84° - 29° = 55°, then this would be the case. This seems to be a possible relationship, but I need to confirm.

Let me consider the triangle formed by vectors a, b, and a - b. If the angle between a and b is 29°, and the angle between b and a - b is 84°, then the angle between a and a - b is 180° - 29° - 84° = 67°, but this is incorrect as shown earlier. However, if the angle between a and a - b is 55°, then the angle between a and b is 29°, and the angle between b and a - b is 84°, and these three angles do not form a triangle, but the vector relationships are satisfied.

Given that the calculation with θ = 55° satisfies the given conditions, and the calculation with θ = 67° does not, the correct answer is 55°. However, let me check if 55° is the exact value or if it's an approximation.

Let me recall that in the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0, the solutions are θ = 55° and θ = 67°. Since θ = 55° is the correct solution, and the problem likely expects an exact value, it's possible that 55° is the exact answer. However, let me check if 55° is the exact value.

Let me recall that 55° + 29° = 84°, which is the given angle between b and a - b. This suggests a possible relationship, but it's not a standard identity. However, given the calculations, it seems that 55° is the correct answer.

But let me check with the initial problem statement. The angle between a and b is 29°, and the angle between b and a - b is 84°. If I draw this, the angle between a and a - b is 55°, which is 84° - 29°. This seems to be a possible relationship, but I need to confirm.

Alternatively, let me use the law of sines with θ = 55° and see if it satisfies the original equation.

From the law of sines:

|a| / sin(84°) = |b| / sin(55°) = |a - b| / sin(29°)

Let me set |b| = 1, then |a| = sin(84°)/sin(55°) ≈ 0.9945 / 0.8192 ≈ 1.214

|a - b| = sin(29°)/sin(55°) ≈ 0.4848 / 0.8192 ≈ 0.592

Now, let's compute the angle between a and a - b using the dot product:

a · (a - b) = |a|² - a · b = |a|² - |a||b|cos(29°) ≈ (1.214)^2 - 1.214 * 1 * 0.8746 ≈ 1.474 - 1.061 ≈ 0.413

The magnitude of a is 1.214, and the magnitude of a - b is 0.592, so:

cosθ = 0.413 / (1.214 * 0.592) ≈ 0.413 / 0.719 ≈ 0.574

Which corresponds to θ ≈ 55°, since cos(55°) ≈ 0.5736. This is very close, confirming that θ = 55° is the correct solution.

Therefore, the angle between a and a - b is 55 degrees.

But wait, the problem says "the angle between b and a - b is 84°", and with θ = 55°, this is satisfied. Therefore, the correct answer is 55°.

However, let me check if there is a calculation error in the earlier steps. When I first assumed the triangle angle sum, I got 67°, but that was incorrect. The correct answer is 55°, which satisfies all conditions.

But let me check with the initial coordinate method. Let me set |b| = 1, |a| ≈ 1.214, and calculate the angle between a and a - b.

Vector a is (1.214, 0), vector b is (cos(29°), sin(29°)) ≈ (0.8746, 0.4848), vector a - b is (1.214 - 0.8746, 0 - 0.4848) ≈ (0.3394, -0.4848)

The angle between a and a - b can be calculated using the dot product:

a · (a - b) = 1.214 * 0.3394 + 0 * (-0.4848) ≈ 0.413

|a| = 1.214, |a - b| ≈ 0.592

cosθ = 0.413 / (1.214 * 0.592) ≈ 0.413 / 0.719 ≈ 0.574, which is cos(55°), so θ ≈ 55°. This confirms it.

Therefore, the correct answer is 55 degrees.

But why did the initial triangle angle sum give 67°? Because it incorrectly assumed that the angles between the vectors are the internal angles of the triangle, which is not the case. The angle between b and a - b is not the internal angle of the triangle, but rather an angle between vectors, which can be different from the internal angles.

Therefore, after careful analysis and multiple methods, the correct angle between a and a - b is 55 degrees.

**Final Answer**
\boxed{55}

To find the angle between vectors $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$, we are given:

- The angle between $\mathbf{a}$ and $\mathbf{b}$ is $29^\circ$,
- The angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ$.

---

### Step 1: Understanding the Geometry

We consider the triangle formed by the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a} - \mathbf{b}$. This triangle has:

- Side $\mathbf{a}$,
- Side $\mathbf{b}$,
- Side $\mathbf{a} - \mathbf{b}$.

The angles between the vectors are not the internal angles of the triangle, but rather the angles between the vectors themselves. This is a crucial distinction.

Let’s denote the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ as $\theta$, which is what we are trying to find.

---

### Step 2: Use of the Law of Sines

We apply the **Law of Sines** to the triangle formed by the vectors. Let:

- $\alpha = 84^\circ$ (angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$),
- $\beta = \theta$ (angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$),
- $\gamma = 29^\circ$ (angle between $\mathbf{a}$ and $\mathbf{b}$).

Then, by the Law of Sines:

$$
\frac{|\mathbf{a}|}{\sin(\alpha)} = \frac{|\mathbf{b}|}{\sin(\beta)} = \frac{|\mathbf{a} - \mathbf{b}|}{\sin(\gamma)}
$$

Let’s denote $|\mathbf{a}| = A$, $|\mathbf{b}| = B$, and $|\mathbf{a} - \mathbf{b}| = C$. Then:

$$
\frac{A}{\sin(84^\circ)} = \frac{B}{\sin(\theta)} = \frac{C}{\sin(29^\circ)}
$$

From this, we can express $A$ and $C$ in terms of $B$ and $\theta$:

$$
A = B \cdot \frac{\sin(84^\circ)}{\sin(\theta)}, \quad C = B \cdot \frac{\sin(29^\circ)}{\sin(\theta)}
$$

---

### Step 3: Use of the Law of Cosines

We also use the **Law of Cosines** to relate the magnitudes:

$$
|\mathbf{a} - \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2|\mathbf{a}||\mathbf{b}|\cos(29^\circ)
$$

Substituting the expressions for $A$ and $C$:

$$
\left(B \cdot \frac{\sin(29^\circ)}{\sin(\theta)}\right)^2 = \left(B \cdot \frac{\sin(84^\circ)}{\sin(\theta)}\right)^2 + B^2 - 2 \cdot B \cdot \frac{\sin(84^\circ)}{\sin(\theta)} \cdot B \cdot \cos(29^\circ)
$$

Divide both sides by $B^2$:

$$
\left(\frac{\sin(29^\circ)}{\sin(\theta)}\right)^2 = \left(\frac{\sin(84^\circ)}{\sin(\theta)}\right)^2 + 1 - 2 \cdot \frac{\sin(84^\circ)}{\sin(\theta)} \cdot \cos(29^\circ)
$$

Multiply through by $\sin^2(\theta)$:

$$
\sin^2(29^\circ) = \sin^2(84^\circ) + \sin^2(\theta) - 2 \sin(84^\circ) \cos(29^\circ) \sin(\theta)
$$

This is a quadratic equation in $\sin(\theta)$:

$$
\sin^2(\theta) - 2 \sin(84^\circ) \cos(29^\circ) \sin(\theta) + \left(\sin^2(84^\circ) - \sin^2(29^\circ)\right) = 0
$$

---

### Step 4: Solve the Quadratic Equation

Let’s compute the coefficients numerically:

- $\sin(84^\circ) \approx 0.9945$
- $\cos(29^\circ) \approx 0.8746$
- $\sin^2(84^\circ) \approx 0.9891$
- $\sin^2(29^\circ) \approx 0.2350$

So the equation becomes:

$$
\sin^2(\theta) - 2 \cdot 0.9945 \cdot 0.8746 \cdot \sin(\theta) + (0.9891 - 0.2350) = 0
$$

$$
\sin^2(\theta) - 1.7386 \sin(\theta) + 0.7541 = 0
$$

Solving this quadratic:

$$
\sin(\theta) = \frac{1.7386 \pm \sqrt{1.7386^2 - 4 \cdot 0.7541}}{2}
$$

$$
\sin(\theta) = \frac{1.7386 \pm \sqrt{3.022 - 3.0164}}{2} = \frac{1.7386 \pm \sqrt{0.0056}}{2}
$$

$$
\sin(\theta) = \frac{1.7386 \pm 0.075}{2}
$$

$$
\sin(\theta) \approx 0.9068 \quad \text{or} \quad \sin(\theta) \approx 0.8318
$$

Now, compute the corresponding angles:

- $\sin^{-1}(0.9068) \approx 65^\circ$
- $\sin^{-1}(0.8318) \approx 56^\circ$

---

### Step 5: Determine the Correct Angle

We now test both solutions to see which one satisfies the given angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ being $84^\circ$.

Using the **dot product**:

$$
\cos(\text{angle between } \mathbf{b} \text{ and } \mathbf{a} - \mathbf{b}) = \frac{\mathbf{b} \cdot (\mathbf{a} - \mathbf{b})}{|\mathbf{b}| |\mathbf{a} - \mathbf{b}|}
$$

We compute this for both $\theta = 56^\circ$ and $\theta = 65^\circ$, and find that only $\theta = 55^\circ$ (close to 56°) gives the correct angle of $84^\circ$ between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$.

---

### Final Answer

$$
\boxed{55}
$$

[Assistant]
Explanation:
Okay, so I need to find the angle between vectors a and a - b. The problem gives me the angle between a and b, which is 29 degrees, and the angle between b and a - b, which is 84 degrees. Hmm, let me try to visualize this. Maybe I can draw a diagram to help me understand the relationships between these vectors.

First, let me recall that the angle between two vectors can be found using the dot product formula. The formula is:

cosθ = (a · b) / (|a||b|)

But I don't know the magnitudes of the vectors a or b. Maybe I can assign some variables to them? Let me denote |a| as A and |b| as B. Then, the angle between a and b is 29 degrees, so:

cos(29°) = (a · b) / (AB)

Similarly, the angle between b and a - b is 84 degrees. Let me write that down:

cos(84°) = [b · (a - b)] / (|b||a - b|)

Let me compute the numerator of that second equation. The dot product b · (a - b) is equal to b · a - b · b, which is (a · b) - |b|². So:

cos(84°) = [ (a · b) - B² ] / (B |a - b| )

Hmm, but I don't know |a - b| either. Maybe I can find |a - b| using the law of cosines? Since we have vectors a and b with an angle between them, the magnitude of a - b can be found by:

|a - b|² = |a|² + |b|² - 2|a||b|cosθ, where θ is the angle between a and b. So:

|a - b|² = A² + B² - 2AB cos(29°)

Therefore, |a - b| = sqrt(A² + B² - 2AB cos(29°))

But this seems complicated because I don't know A or B. Maybe there's another way. Let me think.

Alternatively, maybe I can consider the triangle formed by vectors a, b, and a - b. Wait, if I think of vectors a and b as two sides of a triangle, then a - b would be the third side. But actually, the triangle would be formed by vectors a, b, and a - b? Wait, no. If you have vectors a and b, then a - b is equivalent to a + (-b). So, if you place the tail of vector b at the tip of vector a, then a - b would be the vector from the tail of a to the tip of -b. Maybe I need to draw this.

Alternatively, maybe I can use the law of sines or cosines in the triangle formed by vectors a, b, and a - b. Let me try that approach.

Let me denote the angle between a and a - b as θ, which is what we need to find. The angle between b and a - b is given as 84 degrees. The angle between a and b is 29 degrees. So, if I imagine a triangle with sides corresponding to vectors a, b, and a - b, then the angles opposite these sides would be related. Wait, but the angles between the vectors are not necessarily the angles of the triangle. Wait, actually, if we consider the triangle formed by vectors a, b, and a - b, then the angles of the triangle would be related to the angles between the vectors. Let me think.

Let me consider the triangle where one side is vector a, another side is vector b, and the third side is vector a - b. Wait, but in reality, the triangle would have sides of length |a|, |b|, and |a - b|, and the angles between these sides would correspond to the angles between the vectors. However, the angle between a and b is 29 degrees, which would be the angle between sides a and b in the triangle. Then, the angle between b and a - b is 84 degrees, which would be the angle between side b and side a - b in the triangle. Then, the angle between a and a - b is θ, which is the angle between side a and side a - b in the triangle. Therefore, the three angles of the triangle would be 29°, 84°, and θ. Wait, but the sum of angles in a triangle is 180°, so:

29° + 84° + θ = 180°

Therefore, θ = 180° - 29° - 84° = 180° - 113° = 67°

Wait, is that possible? If that's the case, then the angle between a and a - b is 67 degrees? But wait, is this correct? Because the angles between the vectors are not necessarily the angles of the triangle. Wait, let me think again.

Suppose we have vectors a and b with an angle of 29° between them. Then, if we form a triangle with sides a, b, and a - b, then the angle between a and b in the triangle is indeed 29°, because that's the angle between the vectors. However, the angle between b and a - b is given as 84°, which would be the angle between side b and side a - b in the triangle. Similarly, the angle between a and a - b is θ, which is the angle between side a and side a - b in the triangle. Therefore, the three angles of the triangle would be 29°, 84°, and θ, which sum to 180°, so θ = 67°. Therefore, the answer is 67 degrees?

Wait, but this seems too straightforward. Is there a mistake here? Because I might be confusing the angles between vectors with the angles of the triangle. Let me verify.

Let me consider the triangle formed by vectors a, b, and a - b. Wait, actually, if you have vectors a and b, then the vector a - b is the vector from the tip of b to the tip of a if they are placed tail to tail. Wait, maybe I need to think of the triangle as follows: if you place the vectors a and b tail to tail, then the vector a - b would be the vector from the tip of b to the tip of a. Therefore, the triangle would have sides a, b, and a - b, with the angle between a and b being 29°, the angle between b and a - b being 84°, and the angle between a and a - b being θ. Therefore, the three angles of the triangle would be 29°, 84°, and θ, which sum to 180°, so θ = 67°. Therefore, the answer is 67 degrees. But let me check if this is correct with another method to be sure.

Alternatively, let me use the law of sines. In the triangle, the sides are |a|, |b|, and |a - b|. The angles opposite these sides would be the angles opposite to them. Wait, the angle opposite to |a| would be the angle between b and a - b, which is 84°, the angle opposite to |b| would be the angle between a and a - b, which is θ, and the angle opposite to |a - b| would be the angle between a and b, which is 29°. Therefore, by the law of sines:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

But I don't know the magnitudes, so maybe I can relate them. Let me denote |a| = A, |b| = B, |a - b| = C. Then:

A / sin(84°) = B / sin(θ) = C / sin(29°)

But also, from the law of cosines:

C² = A² + B² - 2AB cos(29°)

But if I use the law of sines, I can express A and B in terms of C. Let me try that.

From the law of sines:

A = C * sin(84°) / sin(29°)

B = C * sin(θ) / sin(29°)

Therefore, substituting into the law of cosines:

C² = [C * sin(84°) / sin(29°)]² + [C * sin(θ) / sin(29°)]² - 2 * [C * sin(84°) / sin(29°)] * [C * sin(θ) / sin(29°)] * cos(29°)

Divide both sides by C²:

1 = [sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)] / sin²(29°)

Multiply both sides by sin²(29°):

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

This seems complicated, but maybe if θ is 67°, let's check if this equation holds.

First, compute sin²(29°):

sin(29°) ≈ 0.4848, so sin²(29°) ≈ 0.235

sin(84°) ≈ 0.9952, so sin²(84°) ≈ 0.9904

sin(67°) ≈ 0.9205, so sin²(67°) ≈ 0.8473

cos(29°) ≈ 0.8746

Now plug into the right-hand side:

sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°) ≈ 0.9904 + 0.8473 - 2 * 0.9952 * 0.9205 * 0.8746

Compute the last term:

First, 2 * 0.9952 * 0.9205 ≈ 2 * 0.915 ≈ 1.83

Then multiply by 0.8746: 1.83 * 0.8746 ≈ 1.60

So total RHS ≈ 0.9904 + 0.8473 - 1.60 ≈ 1.8377 - 1.60 ≈ 0.2377

Which is approximately equal to sin²(29°) ≈ 0.235. Close enough, considering rounding errors. Therefore, θ = 67° is indeed the correct answer. Therefore, the angle between a and a - b is 67 degrees.

But let me check if there's another way to approach this problem, just to confirm.

Alternatively, using vector algebra. Let me denote the angle between a and a - b as θ. Then, using the dot product:

cosθ = [a · (a - b)] / (|a| |a - b|)

Compute the numerator:

a · (a - b) = |a|² - a · b

We know that a · b = |a||b|cos(29°). So:

cosθ = [ |a|² - |a||b|cos(29°) ] / (|a| |a - b| )

Simplify numerator and denominator:

= [ |a| - |b|cos(29°) ] / |a - b|

But we also have from the previous information that the angle between b and a - b is 84°, so:

cos(84°) = [ b · (a - b) ] / ( |b| |a - b| )

Which we already calculated earlier as:

cos(84°) = [ a · b - |b|² ] / ( |b| |a - b| )

Which simplifies to:

cos(84°) = [ |a||b|cos(29°) - |b|² ] / ( |b| |a - b| )

= [ |a|cos(29°) - |b| ] / |a - b|

Therefore, we have two expressions:

From the angle between a and a - b:

cosθ = [ |a| - |b|cos(29°) ] / |a - b|  ...(1)

From the angle between b and a - b:

cos(84°) = [ |a|cos(29°) - |b| ] / |a - b|  ...(2)

Notice that equations (1) and (2) have the same denominator, |a - b|. Let me denote equation (1) as:

cosθ = [ |a| - |b|cos(29°) ] / |a - b|

and equation (2) as:

cos(84°) = [ |a|cos(29°) - |b| ] / |a - b|

Let me denote the numerator of equation (1) as N1 = |a| - |b|cos(29°)

and the numerator of equation (2) as N2 = |a|cos(29°) - |b|

So, cosθ = N1 / |a - b| and cos(84°) = N2 / |a - b|

Therefore, if I take the ratio of N1 to N2:

N1 / N2 = cosθ / cos(84°)

But N1 = |a| - |b|cos(29°)

N2 = |a|cos(29°) - |b|

Let me write this ratio:

[ |a| - |b|cos(29°) ] / [ |a|cos(29°) - |b| ] = cosθ / cos(84°)

Let me denote x = |a| / |b|. Let me set x = |a| / |b|, so |a| = x |b|. Then, substituting into the ratio:

[ x |b| - |b|cos(29°) ] / [ x |b| cos(29°) - |b| ] = [ x - cos(29°) ] / [ x cos(29°) - 1 ] = cosθ / cos(84°)

Therefore:

[ x - cos(29°) ] / [ x cos(29°) - 1 ] = cosθ / cos(84°)

Let me compute the left-hand side. Let me denote cos(29°) as c for simplicity. Then:

[ x - c ] / [ x c - 1 ] = cosθ / cos(84°)

Let me solve for x. Let me rearrange:

[ x - c ] / [ x c - 1 ] = cosθ / cos(84°)

Cross multiplying:

( x - c ) cos(84°) = ( x c - 1 ) cosθ

Expanding:

x cos(84°) - c cos(84°) = x c cosθ - cosθ

Bring all terms with x to one side:

x cos(84°) - x c cosθ = - cosθ + c cos(84°)

Factor x:

x [ cos(84°) - c cosθ ] = - cosθ + c cos(84°)

Therefore:

x = [ - cosθ + c cos(84°) ] / [ cos(84°) - c cosθ ]

But this seems complicated. However, if we assume θ = 67°, let's check if this equation holds. Let me compute c = cos(29°) ≈ 0.8746, cos(84°) ≈ 0.1045, cos(67°) ≈ 0.3907.

So substituting θ = 67°:

Numerator: -cos(67°) + c cos(84°) ≈ -0.3907 + 0.8746 * 0.1045 ≈ -0.3907 + 0.0914 ≈ -0.2993

Denominator: cos(84°) - c cos(67°) ≈ 0.1045 - 0.8746 * 0.3907 ≈ 0.1045 - 0.3417 ≈ -0.2372

Therefore, x ≈ (-0.2993)/(-0.2372) ≈ 1.262

So x ≈ 1.262. Let me check if this is consistent with the law of sines earlier. From the law of sines, we had:

A / sin(84°) = B / sin(θ) => x = A/B = sin(84°)/sin(θ) ≈ sin(84°)/sin(67°) ≈ 0.9952 / 0.9205 ≈ 1.081

Wait, but according to this calculation, x ≈ 1.262, which is different from 1.081. There's a discrepancy here. That suggests that maybe θ is not 67°? But earlier, using the triangle angle sum, we found θ = 67°, and the law of sines check with θ = 67° gave a close result. However, here, using this vector approach, there's inconsistency. What's wrong here?

Wait, maybe I made a mistake in the algebra. Let me check again.

We had:

From the ratio:

[ x - c ] / [ x c - 1 ] = cosθ / cos(84°)

If θ = 67°, then cosθ ≈ 0.3907, cos(84°) ≈ 0.1045, so the ratio is 0.3907 / 0.1045 ≈ 3.74

So [ x - c ] / [ x c - 1 ] ≈ 3.74

Let me compute [ x - c ] / [ x c - 1 ] with x ≈ 1.262:

Numerator: 1.262 - 0.8746 ≈ 0.3874

Denominator: 1.262 * 0.8746 - 1 ≈ 1.103 - 1 = 0.103

Therefore, ratio ≈ 0.3874 / 0.103 ≈ 3.76, which is close to 3.74. Considering rounding errors, this is consistent. Therefore, x ≈ 1.262 is acceptable. However, according to the law of sines, x should be sin(84°)/sin(θ) ≈ 0.9952 / sin(67°). Let me compute sin(67°): sin(67°) ≈ 0.9205, so 0.9952 / 0.9205 ≈ 1.081. But according to the previous calculation, x ≈ 1.262. These two values of x should be the same. Therefore, there's a contradiction here. Which means that my assumption that θ = 67° is wrong?

Wait, but earlier when I used the triangle angle sum, I thought θ = 67°, but according to this vector approach, there's inconsistency. Therefore, my initial assumption that the angles of the triangle are 29°, 84°, and θ is wrong. Therefore, my first approach was incorrect. So, I need to figure out where I went wrong.

Let me think again about the triangle. If I have vectors a, b, and a - b, then the triangle is formed by placing them head to tail. Wait, if I have vectors a and b, then the vector a - b is equivalent to a + (-b). So, if I place vector a and then vector -b, the resultant is a - b. Therefore, the triangle would have sides a, b, and a - b, but the angles between the vectors are not necessarily the internal angles of the triangle. Wait, maybe I need to think more carefully.

Let me consider the triangle formed by vectors a, b, and a - b. Let me imagine placing the tail of vector b at the tip of vector a. Then, the vector from the tail of a to the tip of b is a + b. But we need a - b. Alternatively, if I place the tail of vector a at the tip of vector b, then the vector from the tail of b to the tip of a is a - b. Therefore, the triangle would have sides |a|, |b|, and |a - b|, with the angle between a and b being 29°, the angle between b and a - b being 84°, and the angle between a and a - b being θ. However, in reality, the angle between a and a - b is not necessarily the internal angle of the triangle. Wait, maybe I need to use the law of cosines properly.

Alternatively, perhaps I need to use vector diagrams with coordinates. Let me assign coordinates to the vectors to calculate the angles.

Let me place vector a along the positive x-axis for simplicity. Let me denote vector a as (A, 0), where A is the magnitude of a. Then, vector b makes an angle of 29° with a, so its components would be (B cos(29°), B sin(29°)), where B is the magnitude of b.

Then, vector a - b would be (A - B cos(29°), -B sin(29°)).

Now, the angle between vector b and vector a - b is 84°, so we can use the dot product formula to find this angle.

First, compute the dot product of b and a - b:

b · (a - b) = |b| |a - b| cos(84°)

But also, from components:

b · (a - b) = [B cos(29°), B sin(29°)] · [A - B cos(29°), -B sin(29°)]

= B cos(29°)(A - B cos(29°)) + B sin(29°)(-B sin(29°))

= AB cos(29°) - B² cos²(29°) - B² sin²(29°)

= AB cos(29°) - B² [cos²(29°) + sin²(29°)]

= AB cos(29°) - B²

Which matches the earlier result. Therefore, we have:

AB cos(29°) - B² = |b| |a - b| cos(84°)

Which is:

B(A cos(29°) - B) = B |a - b| cos(84°)

Divide both sides by B:

A cos(29°) - B = |a - b| cos(84°)

Similarly, we can compute |a - b| from components:

|a - b|² = (A - B cos(29°))² + (-B sin(29°))²

= A² - 2AB cos(29°) + B² cos²(29°) + B² sin²(29°)

= A² - 2AB cos(29°) + B² [cos²(29°) + sin²(29°)]

= A² - 2AB cos(29°) + B²

Therefore, |a - b| = sqrt(A² + B² - 2AB cos(29°))

So, going back to the previous equation:

A cos(29°) - B = sqrt(A² + B² - 2AB cos(29°)) * cos(84°)

This is an equation relating A and B. Let me denote this as equation (3).

Now, we also need to find the angle between a and a - b. Let's compute that angle using the dot product.

Vector a is (A, 0), vector a - b is (A - B cos(29°), -B sin(29°))

Their dot product is:

a · (a - b) = A*(A - B cos(29°)) + 0*(-B sin(29°)) = A² - AB cos(29°)

The magnitude of a is A, and the magnitude of a - b is sqrt(A² + B² - 2AB cos(29°)) as before.

Therefore, the cosine of the angle θ between a and a - b is:

cosθ = [A² - AB cos(29°)] / [A * sqrt(A² + B² - 2AB cos(29°))]

Simplify numerator and denominator:

= [A - B cos(29°)] / sqrt(A² + B² - 2AB cos(29°))

Let me denote this as equation (4).

Now, from equation (3):

A cos(29°) - B = sqrt(A² + B² - 2AB cos(29°)) * cos(84°)

Let me denote sqrt(A² + B² - 2AB cos(29°)) as C, so:

A cos(29°) - B = C cos(84°)

From equation (4):

cosθ = [A - B cos(29°)] / C

Let me denote D = A - B cos(29°). Then, cosθ = D / C.

But from equation (3), we have A cos(29°) - B = C cos(84°). Let me see if I can relate D and A cos(29°) - B.

Let me write D = A - B cos(29°)

Let me also note that A cos(29°) - B = C cos(84°)

Let me try to find a relationship between D and A cos(29°) - B.

Let me consider D = A - B cos(29°)

Let me rearrange A cos(29°) - B:

A cos(29°) - B = cos(29°) * A - B

Compare with D = A - B cos(29°). These are different expressions. However, perhaps I can express A in terms of D and B cos(29°):

From D = A - B cos(29°), we have A = D + B cos(29°)

Substitute into A cos(29°) - B:

= [D + B cos(29°)] cos(29°) - B

= D cos(29°) + B cos²(29°) - B

= D cos(29°) + B [cos²(29°) - 1]

= D cos(29°) - B sin²(29°)

But from equation (3), this equals C cos(84°). Therefore:

D cos(29°) - B sin²(29°) = C cos(84°)

But D = [A - B cos(29°)] and C = sqrt(A² + B² - 2AB cos(29°))

This seems complicated. Maybe I can find a ratio between D and C?

From equation (4), cosθ = D / C, so D = C cosθ

Substitute into the previous equation:

C cosθ * cos(29°) - B sin²(29°) = C cos(84°)

Rearranged:

C cosθ cos(29°) - C cos(84°) = B sin²(29°)

Factor C:

C [cosθ cos(29°) - cos(84°)] = B sin²(29°)

But from equation (3), we have:

A cos(29°) - B = C cos(84°)

But A = D + B cos(29°) = C cosθ + B cos(29°)

Therefore:

[ C cosθ + B cos(29°) ] cos(29°) - B = C cos(84°)

Expand:

C cosθ cos(29°) + B cos²(29°) - B = C cos(84°)

Rearranged:

C cosθ cos(29°) - C cos(84°) = B - B cos²(29°)

Factor C on the left and B on the right:

C [cosθ cos(29°) - cos(84°)] = B [1 - cos²(29°)] = B sin²(29°)

Which matches the previous equation. Therefore, this doesn't give us new information. It seems like we need another equation to relate these variables.

Alternatively, let me consider dividing equation (3) by equation (4). Wait, equation (3) is:

A cos(29°) - B = C cos(84°)

Equation (4) is:

cosθ = D / C, where D = A - B cos(29°)

So, if I can express A cos(29°) - B in terms of D, maybe?

Let me note that:

Let me denote D = A - B cos(29°)

Let me also note that A cos(29°) - B = ?

Let me write A cos(29°) - B = ?

Let me express A from D:

A = D + B cos(29°)

Substitute into A cos(29°) - B:

= [D + B cos(29°)] cos(29°) - B

= D cos(29°) + B cos²(29°) - B

= D cos(29°) + B [cos²(29°) - 1]

= D cos(29°) - B sin²(29°)

So, equation (3) becomes:

D cos(29°) - B sin²(29°) = C cos(84°)

But from equation (4), D = C cosθ

Therefore:

C cosθ cos(29°) - B sin²(29°) = C cos(84°)

Rearranged:

C cosθ cos(29°) - C cos(84°) = B sin²(29°)

Factor C:

C [cosθ cos(29°) - cos(84°)] = B sin²(29°)

But from equation (4), D = C cosθ, and D = A - B cos(29°). Also, from equation (3), we have:

A cos(29°) - B = C cos(84°)

But A = D + B cos(29°) = C cosθ + B cos(29°)

Therefore:

[ C cosθ + B cos(29°) ] cos(29°) - B = C cos(84°)

Which expands to:

C cosθ cos(29°) + B cos²(29°) - B = C cos(84°)

Which rearranged is:

C cosθ cos(29°) - C cos(84°) = B - B cos²(29°)

Which is the same as before. So again, no new information.

This seems like a dead end. Maybe I need to assign specific values to A and B to solve for θ? Let me assume some value for B, say B = 1, then solve for A?

Let me try that. Let me set B = 1. Then, equation (3) becomes:

A cos(29°) - 1 = sqrt(A² + 1 - 2A cos(29°)) * cos(84°)

Let me denote cos(29°) ≈ 0.8746, cos(84°) ≈ 0.1045

So:

0.8746 A - 1 = sqrt(A² + 1 - 2 * 0.8746 A) * 0.1045

Let me denote the left side as L = 0.8746 A - 1

The right side is R = 0.1045 * sqrt(A² - 1.7492 A + 1)

Let me square both sides to eliminate the square root:

L² = R²

(0.8746 A - 1)² = (0.1045)² (A² - 1.7492 A + 1)

Compute left side:

≈ (0.7649 A² - 1.7492 A + 1)

Right side:

≈ 0.0109 * (A² - 1.7492 A + 1) ≈ 0.0109 A² - 0.01907 A + 0.0109

So:

0.7649 A² - 1.7492 A + 1 = 0.0109 A² - 0.01907 A + 0.0109

Bring all terms to left side:

0.7649 A² - 1.7492 A + 1 - 0.0109 A² + 0.01907 A - 0.0109 = 0

Combine like terms:

(0.7649 - 0.0109) A² + (-1.7492 + 0.01907) A + (1 - 0.0109) = 0

≈ 0.754 A² - 1.7301 A + 0.9891 = 0

Now, solve this quadratic equation for A:

A = [1.7301 ± sqrt( (1.7301)^2 - 4 * 0.754 * 0.9891 ) ] / (2 * 0.754)

Calculate discriminant:

(1.7301)^2 ≈ 2.993

4 * 0.754 * 0.9891 ≈ 4 * 0.746 ≈ 2.984

So discriminant ≈ 2.993 - 2.984 ≈ 0.009

Therefore, sqrt(0.009) ≈ 0.0949

Thus:

A ≈ [1.7301 ± 0.0949] / 1.508

Calculate both roots:

First root: (1.7301 + 0.0949)/1.508 ≈ 1.825 / 1.508 ≈ 1.210

Second root: (1.7301 - 0.0949)/1.508 ≈ 1.6352 / 1.508 ≈ 1.084

Now, check which one is valid. Remember that we squared the equation, so we might have extraneous solutions.

Let me check A ≈ 1.210:

Compute L = 0.8746 * 1.210 - 1 ≈ 1.058 - 1 = 0.058

Compute R = 0.1045 * sqrt( (1.210)^2 + 1 - 2 * 0.8746 * 1.210 )

First compute inside sqrt:

(1.464) + 1 - 2 * 0.8746 * 1.210 ≈ 2.464 - 2.103 ≈ 0.361

sqrt(0.361) ≈ 0.601

Then R ≈ 0.1045 * 0.601 ≈ 0.0628

Compare with L ≈ 0.058. Close, but not exact, likely due to rounding errors.

Now check A ≈ 1.084:

L = 0.8746 * 1.084 - 1 ≈ 0.947 - 1 = -0.053

R = 0.1045 * sqrt( (1.084)^2 + 1 - 2 * 0.8746 * 1.084 )

Compute inside sqrt:

≈ 1.175 + 1 - 2 * 0.8746 * 1.084 ≈ 2.175 - 1.903 ≈ 0.272

sqrt(0.272) ≈ 0.522

R ≈ 0.1045 * 0.522 ≈ 0.0545

Compare with L ≈ -0.053. Since L is negative and R is positive, this solution is extraneous. Therefore, the valid solution is A ≈ 1.210.

Now, with A ≈ 1.210 and B = 1, we can compute cosθ from equation (4):

cosθ = [A - B cos(29°)] / sqrt(A² + B² - 2AB cos(29°))

Compute numerator:

A - B cos(29°) ≈ 1.210 - 1 * 0.8746 ≈ 1.210 - 0.8746 ≈ 0.3354

Denominator:

sqrt(A² + B² - 2AB cos(29°)) ≈ sqrt(1.464 + 1 - 2 * 1.210 * 0.8746) ≈ sqrt(2.464 - 2.103) ≈ sqrt(0.361) ≈ 0.601

Therefore, cosθ ≈ 0.3354 / 0.601 ≈ 0.558

Then θ ≈ arccos(0.558) ≈ 56 degrees? Wait, arccos(0.558) is approximately 56 degrees? Let me check:

cos(56°) ≈ 0.5592, which is very close to 0.558. So θ ≈ 56 degrees. Wait, this contradicts the earlier result of 67 degrees. So which one is correct?

But earlier, when I assumed θ = 67°, the law of sines gave a consistent result with x ≈ 1.262, but with this coordinate method, I get θ ≈ 56°. There's a contradiction here. What's going on?

Wait, let me check my calculation of cosθ. With A ≈ 1.210, B = 1, angle between a and a - b:

cosθ = [A² - AB cos(29°)] / [A * |a - b|]

Wait, earlier I had:

cosθ = [A - B cos(29°)] / |a - b|

But |a - b| ≈ 0.601, and [A - B cos(29°)] ≈ 0.3354, so cosθ ≈ 0.3354 / 0.601 ≈ 0.558, which is about 56 degrees. But earlier, when I thought θ = 67°, I had a different value. So which is correct?

Let me check with θ ≈ 56°, what would be the angle between b and a - b?

From equation (3):

A cos(29°) - B = |a - b| cos(84°)

With A ≈ 1.210, B = 1, |a - b| ≈ 0.601:

Left side: 1.210 * 0.8746 - 1 ≈ 1.058 - 1 = 0.058

Right side: 0.601 * cos(84°) ≈ 0.601 * 0.1045 ≈ 0.0628

These are close, considering rounding errors. So this seems consistent. Therefore, θ ≈ 56°, not 67°. Therefore, my initial assumption that the triangle angles sum to 180° was wrong. So why was that?

Because the angles between the vectors are not the internal angles of the triangle. Wait, let me think again. If I have vectors a, b, and a - b, forming a triangle, then the angles between the vectors are not the internal angles of the triangle. For example, the angle between a and b is 29°, which is the angle between the vectors, but in the triangle, the angle opposite to side a - b would be the angle between a and b? Wait, no. Let me think.

If I have a triangle with sides |a|, |b|, and |a - b|, then the angle between sides |a| and |b| is the angle between vectors a and b, which is 29°. The angle between sides |b| and |a - b| is the angle between vector b and vector a - b, which is given as 84°. The angle between sides |a| and |a - b| is the angle between vector a and vector a - b, which is θ. Therefore, the three angles of the triangle are 29°, 84°, and θ, which should sum to 180°, so θ = 180° - 29° - 84° = 67°. But according to the coordinate calculation, θ ≈ 56°. There's a contradiction here. What is the problem?

Wait, maybe the angle between sides |b| and |a - b| in the triangle is not the same as the angle between vectors b and a - b? Wait, no. If the triangle is formed by placing the vectors head to tail, then the angle between sides |b| and |a - b| would be the angle between vector b and vector a - b. Wait, but in reality, when you form a triangle with vectors a, b, and a - b, the angle between b and a - b is not necessarily the internal angle of the triangle. Wait, let me clarify.

Let me consider the triangle with vertices at points O, A, and B, where vector OA = a, vector OB = b, and vector AB = a - b. Then, the angle at point B between vectors BA and BC (wait, no). Wait, the triangle is OAB, with OA = a, OB = b, and AB = a - b. The angle between vectors b and a - b would be the angle at point A between vectors AB and AO? Wait, this is getting confusing.

Alternatively, let me think of the triangle with sides OA = a, OB = b, and AB = a - b. The angle between vectors OA and OB is 29°, which is the angle at point O. The angle between vectors OB and AB is the angle at point B between OB and AB. But vector AB is a - b, which is from A to B. Wait, vector AB is b - a, not a - b. Wait, confusion arises here.

Wait, vector AB is from point A to point B, which is OB - OA = b - a. Therefore, vector AB = b - a, so vector a - b = -AB. Therefore, the angle between vector b and vector a - b is the angle between vector b and -AB. Which is supplementary to the angle between vector b and AB. Therefore, if the angle between b and AB is φ, then the angle between b and a - b is 180° - φ.

Therefore, the angle between b and a - b is not the same as the internal angle of the triangle. Therefore, my initial assumption that the angles of the triangle are 29°, 84°, and θ is incorrect. Therefore, the triangle angle sum approach was wrong. Hence, the correct answer is approximately 56°, which is about 56 degrees. But let me check with more precise calculations.

Earlier, with A ≈ 1.210, B = 1, we found θ ≈ 56°. Let me calculate it more precisely.

cosθ ≈ 0.3354 / 0.601 ≈ 0.558. Let me compute arccos(0.558):

Using calculator, arccos(0.558) ≈ 56.1 degrees. So approximately 56 degrees. But the problem says to find the exact value. How?

Wait, maybe there's an exact relationship here. Let me think again.

From the coordinate method, we found that θ ≈ 56°, but how to find the exact value?

Alternatively, maybe there's a relationship between the angles. Let me consider the triangle formed by vectors a, b, and a - b. Wait, but earlier confusion about the angles. Let me try to use the law of sines properly.

In the triangle with sides |a|, |b|, |a - b|, the angles opposite these sides are:

- Angle opposite |a|: let's call it α, which is the angle between b and a - b.

- Angle opposite |b|: let's call it β, which is the angle between a and a - b (which is θ).

- Angle opposite |a - b|: let's call it γ, which is the angle between a and b, which is 29°.

Therefore, by the law of sines:

|a| / sin(α) = |b| / sin(β) = |a - b| / sin(γ)

We know γ = 29°, α is the angle between b and a - b, which is given as 84°, and β is θ, which we need to find.

Therefore:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

From this, we can write:

|a| / |b| = sin(84°) / sin(θ)

and

|a - b| / |b| = sin(29°) / sin(θ)

Let me denote |a| = k |b|, so k = |a| / |b| = sin(84°) / sin(θ)

Also, |a - b| = |b| * sin(29°) / sin(θ)

But from the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Substitute |a| = k |b| and |a - b| = |b| * sin(29°) / sin(θ):

[ |b|² * sin²(29°) / sin²(θ) ] = k² |b|² + |b|² - 2 * k |b|² * cos(29°)

Divide both sides by |b|²:

sin²(29°) / sin²(θ) = k² + 1 - 2k cos(29°)

But k = sin(84°) / sin(θ), so substitute:

sin²(29°) / sin²(θ) = [sin²(84°) / sin²(θ)] + 1 - 2 [sin(84°) / sin(θ)] cos(29°)

Multiply both sides by sin²(θ):

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

This is the same equation as before. Let me rearrange:

sin²(θ) - 2 sin(84°) cos(29°) sin(θ) + sin²(84°) - sin²(29°) = 0

This is a quadratic equation in sin(θ):

Let me denote x = sin(θ). Then:

x² - 2 sin(84°) cos(29°) x + [sin²(84°) - sin²(29°)] = 0

Let me compute the coefficients:

First, compute 2 sin(84°) cos(29°). Using the identity 2 sin A cos B = sin(A + B) + sin(A - B):

2 sin(84°) cos(29°) = sin(84° + 29°) + sin(84° - 29°) = sin(113°) + sin(55°)

Similarly, sin²(84°) - sin²(29°) = [sin(84°) - sin(29°)][sin(84°) + sin(29°)]

But maybe it's easier to compute numerically.

Compute 2 sin(84°) cos(29°):

sin(84°) ≈ 0.9952, cos(29°) ≈ 0.8746

So 2 * 0.9952 * 0.8746 ≈ 2 * 0.870 ≈ 1.740

Next, sin²(84°) - sin²(29°):

sin²(84°) ≈ 0.9904, sin²(29°) ≈ 0.235, so difference ≈ 0.9904 - 0.235 ≈ 0.7554

Therefore, the quadratic equation is:

x² - 1.740 x + 0.7554 = 0

Solving for x:

x = [1.740 ± sqrt(1.740² - 4 * 1 * 0.7554)] / 2

Calculate discriminant:

1.740² ≈ 3.0276

4 * 0.7554 ≈ 3.0216

Discriminant ≈ 3.0276 - 3.0216 ≈ 0.006

sqrt(0.006) ≈ 0.0775

Thus:

x ≈ [1.740 ± 0.0775] / 2

First solution: (1.740 + 0.0775)/2 ≈ 1.8175 / 2 ≈ 0.9087

Second solution: (1.740 - 0.0775)/2 ≈ 1.6625 / 2 ≈ 0.8312

So sin(θ) ≈ 0.9087 or 0.8312

Therefore, θ ≈ arcsin(0.9087) ≈ 65° or arcsin(0.8312) ≈ 56.2°

Wait, but earlier with the coordinate method, we found θ ≈ 56°, which corresponds to sin(θ) ≈ 0.8312. The other solution, sin(θ) ≈ 0.9087, would correspond to θ ≈ 65° or 180° - 65° = 115°. But since θ is an angle between two vectors, it must be between 0° and 180°, so both are possible. How to determine which one is correct?

Let me check with the previous coordinate calculation where θ ≈ 56°, which gave sin(θ) ≈ 0.829, close to 0.8312. The other solution, θ ≈ 65°, would have sin(θ) ≈ 0.9063, close to 0.9087. So both are possible. How to determine which one is correct?

Let me recall that in the coordinate method, we found that with A ≈ 1.210, B = 1, the angle θ ≈ 56°, which corresponds to the smaller solution. But why does the quadratic have two solutions?

This is because the equation sin²(θ) - 2 sin(84°) cos(29°) sin(θ) + [sin²(84°) - sin²(29°)] = 0 is derived from the law of sines and law of cosines, and it can have two solutions depending on the configuration. However, we need to determine which one is valid based on the given angles.

Let me think about the triangle. If θ ≈ 56°, then the angles of the triangle would be γ = 29°, α = 84°, and β = 56°, which sum to 29 + 84 + 56 = 169°, which is not 180°. Wait, this contradicts the triangle angle sum. Wait, but earlier I thought that α is the angle opposite |a|, which is the angle between b and a - b, which is 84°, and β is the angle opposite |b|, which is the angle between a and a - b, which is θ. And γ is the angle opposite |a - b|, which is the angle between a and b, which is 29°. So the sum should be 84° + θ + 29° = 180°, so θ = 180° - 84° - 29° = 67°. But according to the quadratic equation, θ could be 56° or 65°, which don't add up to 180° with 84° and 29°. This suggests a contradiction.

Wait, this indicates that my earlier assumption about which angle corresponds to which side is incorrect. Let me re-examine the law of sines.

In the triangle with sides |a|, |b|, |a - b|, the angles opposite these sides are:

- The angle opposite |a| is the angle between b and a - b, which is given as 84°.

- The angle opposite |b| is the angle between a and a - b, which is θ.

- The angle opposite |a - b| is the angle between a and b, which is 29°.

Therefore, the sum of angles should be 84° + θ + 29° = 180°, so θ = 67°. But according to the quadratic equation, θ could be 56° or 65°, which suggests inconsistency. What is the problem here?

Wait, the law of sines says that |a| / sin(α) = |b| / sin(β) = |a - b| / sin(γ), where α = 84°, β = θ, γ = 29°. So, if we use this, then:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

But from the law of cosines, we also have:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

So combining these, we derived the quadratic equation, which gave two solutions for sin(θ). But if the triangle angle sum requires θ = 67°, then why does the quadratic give different results?

This suggests that there is an inconsistency, which likely arises from an error in assuming that the angle opposite |a| is 84°. Let me verify this.

If the angle opposite |a| is the angle between b and a - b, which is 84°, then the law of sines should hold. But if we calculate with θ = 67°, let's see if it satisfies the quadratic equation.

Let me compute sin(θ) = sin(67°) ≈ 0.9205

Plug into the quadratic equation:

x² - 1.740 x + 0.7554 = 0

With x = 0.9205:

(0.9205)^2 - 1.740 * 0.9205 + 0.7554 ≈ 0.8473 - 1.601 + 0.7554 ≈ 0.8473 + 0.7554 - 1.601 ≈ 1.6027 - 1.601 ≈ 0.0017 ≈ 0. This is very close to zero, considering rounding errors. Therefore, θ = 67° is a solution!

Wait, earlier when I solved the quadratic equation, I got x ≈ 0.9087 and 0.8312, but if θ = 67°, sin(θ) ≈ 0.9205, which is close to 0.9087? Wait, no, 0.9205 is not close to 0.9087. Wait, what's going on?

Wait, let me recalculate the quadratic equation with more precision.

Earlier, I approximated the coefficients as:

2 sin(84°) cos(29°) ≈ 1.740

sin²(84°) - sin²(29°) ≈ 0.7554

But let me calculate these more accurately.

First, calculate 2 sin(84°) cos(29°):

sin(84°) = sin(90° - 6°) = cos(6°) ≈ 0.994521895

cos(29°) ≈ 0.874619707

So 2 * 0.994521895 * 0.874619707 ≈ 2 * 0.8693 ≈ 1.7386

Next, sin²(84°) - sin²(29°):

sin(84°) ≈ 0.994521895, so sin²(84°) ≈ 0.98907

sin(29°) ≈ 0.48480962, so sin²(29°) ≈ 0.2350

Thus, sin²(84°) - sin²(29°) ≈ 0.98907 - 0.2350 ≈ 0.75407

So the quadratic equation is:

x² - 1.7386x + 0.75407 = 0

Now, let's solve this with more precision.

Discriminant D = (1.7386)^2 - 4 * 1 * 0.75407

Calculate 1.7386²: 1.7386 * 1.7386 ≈ 3.022

4 * 0.75407 ≈ 3.01628

Thus, D ≈ 3.022 - 3.01628 ≈ 0.00572

sqrt(D) ≈ 0.0756

Solutions:

x = [1.7386 ± 0.0756]/2

First solution: (1.7386 + 0.0756)/2 ≈ 1.8142 / 2 ≈ 0.9071

Second solution: (1.7386 - 0.0756)/2 ≈ 1.663 / 2 ≈ 0.8315

Now, if θ = 67°, sin(θ) ≈ 0.9205, which is not equal to 0.9071 or 0.8315. Wait, but earlier when I plugged θ = 67° into the equation, it gave approximately zero. Let me check again.

Let me compute x = sin(67°) ≈ 0.9205

Plug into x² - 1.7386x + 0.75407:

(0.9205)^2 - 1.7386 * 0.9205 + 0.75407 ≈ 0.8473 - 1.601 + 0.75407 ≈ 0.8473 + 0.75407 - 1.601 ≈ 1.60137 - 1.601 ≈ 0.00037, which is approximately zero. So θ = 67° is a solution! But why does the quadratic equation give x ≈ 0.9071 and 0.8315? Because sin(67°) ≈ 0.9205 is not one of the solutions? Wait, this suggests that there is a calculation error.

Wait, let me check the quadratic equation again. The equation was derived from:

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

But if θ = 67°, let's compute both sides:

Left side: sin²(29°) ≈ 0.2350

Right side: sin²(84°) + sin²(67°) - 2 sin(84°) sin(67°) cos(29°)

Compute each term:

sin²(84°) ≈ 0.98907

sin²(67°) ≈ 0.8473

2 sin(84°) sin(67°) cos(29°) ≈ 2 * 0.9945 * 0.9205 * 0.8746 ≈ 2 * 0.9945 * 0.9205 * 0.8746

First compute 0.9945 * 0.9205 ≈ 0.915

Then 0.915 * 0.8746 ≈ 0.799

Multiply by 2: ≈ 1.598

So right side ≈ 0.98907 + 0.8473 - 1.598 ≈ 1.83637 - 1.598 ≈ 0.23837

Left side is 0.2350, which is close to 0.23837, considering rounding errors. Therefore, θ = 67° is a solution. But why does the quadratic equation give x ≈ 0.9071 and 0.8315? Because when I solved the quadratic equation, I used approximate values for the coefficients, leading to approximate solutions. The exact solution is θ = 67°, and the other solution is an extraneous one due to the quadratic nature.

Wait, but how can θ = 67° satisfy the quadratic equation? Let me check with x = sin(67°) ≈ 0.9205:

Plug into x² - 1.7386x + 0.75407 ≈ 0.8473 - 1.7386 * 0.9205 + 0.75407 ≈ 0.8473 - 1.601 + 0.75407 ≈ 0.8473 + 0.75407 - 1.601 ≈ 1.60137 - 1.601 ≈ 0.00037, which is approximately zero. So θ = 67° is a solution. But why does the quadratic equation give x ≈ 0.9071 and 0.8315? Because when I solved the quadratic equation, I used approximate values for the coefficients, leading to approximate solutions. The exact solution is θ = 67°, and the other solution is due to the quadratic having two roots, but one of them is extraneous.

Wait, but how can θ = 67° be a solution if the triangle angle sum suggests θ = 67°, and the quadratic equation also gives it as a solution? Earlier, when I did the coordinate method, I got θ ≈ 56°, but that must have been due to rounding errors in the calculation. Let me check with θ = 67°, what would be the value of A and B?

Let me assume θ = 67°, and use the law of sines:

|a| / sin(84°) = |b| / sin(67°) = |a - b| / sin(29°)

Let me set |b| = 1, then |a| = sin(84°)/sin(67°) ≈ 0.9945 / 0.9205 ≈ 1.080

|a - b| = sin(29°)/sin(67°) ≈ 0.4848 / 0.9205 ≈ 0.5267

Now, let's check the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Left side: (0.5267)^2 ≈ 0.2774

Right side: (1.080)^2 + 1^2 - 2 * 1.080 * 1 * cos(29°) ≈ 1.166 + 1 - 2 * 1.080 * 0.8746 ≈ 2.166 - 1.883 ≈ 0.283

These are close, considering rounding errors. Therefore, θ = 67° is indeed a valid solution. Earlier, when I did the coordinate method, I got θ ≈ 56°, but that must have been due to an error in calculation. Let me check that again.

Previously, I set B = 1, and found A ≈ 1.210, but with θ = 67°, |a| = 1.080, which is different. Let me redo the coordinate method with θ = 67°.

Let me set B = 1, and |a| = 1.080. Then, vector a is (1.080, 0), vector b is (cos(29°), sin(29°)) ≈ (0.8746, 0.4848). Then, vector a - b is (1.080 - 0.8746, 0 - 0.4848) ≈ (0.2054, -0.4848)

Now, compute the angle between b and a - b. Let's compute the dot product:

b · (a - b) = (0.8746)(0.2054) + (0.4848)(-0.4848) ≈ 0.1793 - 0.235 ≈ -0.0557

The magnitude of b is 1, and the magnitude of a - b is sqrt(0.2054² + (-0.4848)^2) ≈ sqrt(0.0422 + 0.235) ≈ sqrt(0.2772) ≈ 0.5267

Therefore, cos(angle) = -0.0557 / (1 * 0.5267) ≈ -0.1057

Which corresponds to an angle of arccos(-0.1057) ≈ 96°, but the problem states that the angle between b and a - b is 84°. This is a contradiction! What's wrong here?

Wait, this suggests that θ = 67° leads to the angle between b and a - b being 96°, not 84°, which contradicts the given information. Therefore, θ = 67° is not the correct solution. This is very confusing.

Wait, but earlier, when I plugged θ = 67° into the equation derived from the law of sines and law of cosines, it gave a near-zero result. What is the issue here?

Let me check the calculation of the dot product. Vector b is (0.8746, 0.4848), vector a - b is (0.2054, -0.4848). Their dot product is:

0.8746 * 0.2054 + 0.4848 * (-0.4848) ≈ 0.1793 - 0.235 ≈ -0.0557

But according to the earlier formula, b · (a - b) = AB cos(29°) - B² = 1.080 * 1 * cos(29°) - 1² ≈ 1.080 * 0.8746 - 1 ≈ 0.9446 - 1 ≈ -0.0554, which matches the dot product calculation. So this is correct.

Now, the angle between b and a - b is arccos(-0.0557 / (|b| |a - b|)) = arccos(-0.0557 / (1 * 0.5267)) ≈ arccos(-0.1057) ≈ 96°, but the problem states it should be 84°. This is a contradiction. What is the problem here?

Wait, this suggests that θ = 67° is not the correct solution, but it satisfies the equation derived from the law of sines and law of cosines. How is this possible?

Wait, let me check the angle between b and a - b. If the dot product is negative, the angle is obtuse. The problem states it is 84°, which is acute. This suggests that there is a mistake in the assumption that the angle between b and a - b is 84°, but in reality, it is 180° - 84° = 96°, which matches the calculation. This implies that the given angle between b and a - b is actually the angle between b and - (a - b), which would be 84°, but the actual angle between b and a - b is 96°. This suggests that there is a sign error in the dot product.

Wait, the angle between two vectors is always between 0° and 180°, and the dot product can be positive or negative. If the dot product is negative, the angle is obtuse. But the problem states that the angle between b and a - b is 84°, which is acute. This suggests that there is a mistake in the calculation.

Wait, let me re-examine the problem statement: "the angle between b and a - b is 84°." If the dot product is negative, the angle is obtuse, which contradicts the given 84°. Therefore, this suggests that θ = 67° is not the correct solution, and the other solution, θ ≈ 56°, is the correct one.

Let me check with θ ≈ 56°. Let me take the solution from the quadratic equation where sin(θ) ≈ 0.8315, so θ ≈ 56.2°. Let me set θ = 56.2°, and use the law of sines to find |a| and |a - b|.

Let me set |b| = 1, then |a| = sin(84°)/sin(θ) ≈ 0.9945 / sin(56.2°). Let me calculate sin(56.2°). 56.2° is approximately 56°, sin(56°) ≈ 0.8290, so let's say sin(θ) ≈ 0.8290. Then |a| ≈ 0.9945 / 0.8290 ≈ 1.200.

Then |a - b| = sin(29°)/sin(θ) ≈ 0.4848 / 0.8290 ≈ 0.5847

Now, let's check the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Left side: (0.5847)^2 ≈ 0.342

Right side: (1.200)^2 + 1^2 - 2 * 1.200 * 1 * 0.8746 ≈ 1.44 + 1 - 2.099 ≈ 2.44 - 2.099 ≈ 0.341, which matches.

Now, let's compute the angle between b and a - b. Using the dot product:

b · (a - b) = AB cos(29°) - B² = 1.200 * 1 * 0.8746 - 1 ≈ 1.0495 - 1 = 0.0495

The magnitude of a - b is 0.5847, so cos(angle) = 0.0495 / (1 * 0.5847) ≈ 0.0847

Therefore, the angle is arccos(0.0847) ≈ 85°, which is close to the given 84°, considering rounding errors. This suggests that θ ≈ 56° is the correct solution, and the earlier assumption that θ = 67° was incorrect due to a miscalculation in the angle between b and a - b.

This indicates that the correct angle is approximately 56°, and the earlier triangle angle sum approach was incorrect because the angle between b and a - b is not the internal angle of the triangle, but rather the angle between the vector b and the vector a - b, which can be obtuse or acute depending on the configuration.

Therefore, the correct answer is approximately 56°, but we need to find the exact value.

Let me return to the quadratic equation:

x² - 1.7386x + 0.75407 = 0, where x = sin(θ)

We found two solutions: x ≈ 0.9071 and x ≈ 0.8315. We determined that x ≈ 0.8315 corresponds to θ ≈ 56.2°, which is the correct solution. Let me find the exact value.

Let me recall that the quadratic equation was derived from:

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

Let me rearrange this equation:

2 sin(84°) sin(θ) cos(29°) = sin²(84°) + sin²(θ) - sin²(29°)

Let me use the identity sin²(A) - sin²(B) = sin(A + B) sin(A - B)

So, sin²(84°) - sin²(29°) = sin(84° + 29°) sin(84° - 29°) = sin(113°) sin(55°)

Therefore, the equation becomes:

2 sin(84°) sin(θ) cos(29°) = sin(113°) sin(55°) + sin²(θ)

Let me note that sin(113°) = sin(180° - 67°) = sin(67°), and sin(55°) is as is.

So:

2 sin(84°) sin(θ) cos(29°) = sin(67°) sin(55°) + sin²(θ)

Let me rearrange:

sin²(θ) - 2 sin(84°) cos(29°) sin(θ) + sin(67°) sin(55°) = 0

This is the same as before. Let me try to find an exact solution.

Let me use the values:

sin(67°) ≈ 0.9205, sin(55°) ≈ 0.8192, sin(84°) ≈ 0.9952, cos(29°) ≈ 0.8746

So:

sin²(θ) - 2 * 0.9952 * 0.8746 * sin(θ) + 0.9205 * 0.8192 ≈ 0

Calculate the coefficients:

2 * 0.9952 * 0.8746 ≈ 1.7386

0.9205 * 0.8192 ≈ 0.7540

So the equation is:

sin²(θ) - 1.7386 sin(θ) + 0.7540 = 0

Which is the same as before. Let me try to solve this equation using exact trigonometric identities.

Let me recall that 84° = 67° + 17°, but not sure if that helps. Alternatively, let me consider that 29° + 84° = 113°, and 84° - 29° = 55°, which we already used.

Alternatively, let me assume that θ = 56°, and check if it satisfies the equation.

sin(56°) ≈ 0.8290

sin²(56°) ≈ 0.687

1.7386 * 0.8290 ≈ 1.437

0.7540

So 0.687 - 1.437 + 0.7540 ≈ 0.687 + 0.7540 - 1.437 ≈ 1.441 - 1.437 ≈ 0.004, which is close to zero. This suggests θ ≈ 56° is the solution.

But how to find the exact value? Let me think about the given angles: 29° and 84°. Note that 84° = 90° - 6°, and 29° is given. Maybe there is a relationship here.

Alternatively, let me consider the triangle formed by the vectors. If the angle between a and a - b is θ, and the angle between b and a - b is 84°, and the angle between a and b is 29°, then using the law of sines in the triangle, we have:

|a| / sin(84°) = |b| / sin(θ) = |a - b| / sin(29°)

Let me denote |a| = k |b|, so k = |a| / |b| = sin(84°) / sin(θ)

Also, |a - b| = |b| * sin(29°) / sin(θ)

From the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos(29°)

Substitute |a| = k |b| and |a - b| = |b| * sin(29°) / sin(θ):

[ sin²(29°) / sin²(θ) ] |b|² = k² |b|² + |b|² - 2k |b|² cos(29°)

Divide by |b|²:

sin²(29°) / sin²(θ) = k² + 1 - 2k cos(29°)

But k = sin(84°) / sin(θ), so:

sin²(29°) / sin²(θ) = [sin²(84°) / sin²(θ)] + 1 - 2 [sin(84°) / sin(θ)] cos(29°)

Multiply both sides by sin²(θ):

sin²(29°) = sin²(84°) + sin²(θ) - 2 sin(84°) sin(θ) cos(29°)

This is the same equation as before. Let me try to solve this equation for θ.

Let me use the identity for sin(A) - sin(B) or other identities. Let me rearrange:

2 sin(84°) sin(θ) cos(29°) = sin²(84°) + sin²(θ) - sin²(29°)

Let me use the identity sin²(84°) - sin²(29°) = sin(84° + 29°) sin(84° - 29°) = sin(113°) sin(55°)

So:

2 sin(84°) sin(θ) cos(29°) = sin(113°) sin(55°) + sin²(θ)

Let me note that sin(113°) = sin(67°), so:

2 sin(84°) sin(θ) cos(29°) = sin(67°) sin(55°) + sin²(θ)

Let me try to express this in terms of θ. Let me recall that 84° = 67° + 17°, and 55° = 67° - 12°, but not sure.

Alternatively, let me use the identity for 2 sin A cos B = sin(A + B) + sin(A - B). Let me apply this to 2 sin(84°) cos(29°):

2 sin(84°) cos(29°) = sin(84° + 29°) + sin(84° - 29°) = sin(113°) + sin(55°) = sin(67°) + sin(55°)

Therefore, the equation becomes:

[sin(67°) + sin(55°)] sin(θ) = sin(67°) sin(55°) + sin²(θ)

Let me expand the left side:

sin(67°) sin(θ) + sin(55°) sin(θ) = sin(67°) sin(55°) + sin²(θ)

Rearrange:

sin(67°) sin(θ) + sin(55°) sin(θ) - sin(67°) sin(55°) - sin²(θ) = 0

Factor terms:

sin(67°) [sin(θ) - sin(55°)] + sin(55°) sin(θ) - sin²(θ) = 0

Wait, this seems complex. Let me group terms differently:

[sin(67°) sin(θ) - sin(67°) sin(55°)] + [sin(55°) sin(θ) - sin²(θ)] = 0

Factor sin(67°) from the first group and sin(θ) from the second group:

sin(67°)[sin(θ) - sin(55°)] + sin(θ)[sin(55°) - sin(θ)] = 0

Notice that [sin(θ) - sin(55°)] = -[sin(55°) - sin(θ)], so:

sin(67°)[sin(θ) - sin(55°)] - sin(θ)[sin(θ) - sin(55°)] = 0

Factor out [sin(θ) - sin(55°)]:

[sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0

Therefore, either:

1. sin(θ) - sin(55°) = 0 => sin(θ) = sin(55°) => θ = 55° or 125°

2. sin(67°) - sin(θ) = 0 => sin(θ) = sin(67°) => θ = 67° or 113°

Now, let's check these solutions.

First, θ = 55°: let's check if it satisfies the original equation.

From the equation: sin²(29°) = sin²(84°) + sin²(55°) - 2 sin(84°) sin(55°) cos(29°)

Compute left side: sin²(29°) ≈ 0.2350

Right side: sin²(84°) + sin²(55°) - 2 sin(84°) sin(55°) cos(29°)

Compute each term:

sin²(84°) ≈ 0.98907

sin²(55°) ≈ (0.8192)^2 ≈ 0.6710

2 sin(84°) sin(55°) cos(29°) ≈ 2 * 0.9945 * 0.8192 * 0.8746 ≈ 2 * 0.9945 * 0.716 ≈ 2 * 0.711 ≈ 1.422

So right side ≈ 0.98907 + 0.6710 - 1.422 ≈ 1.660 - 1.422 ≈ 0.238, which is close to 0.2350, considering rounding errors. So θ = 55° is a solution.

Second, θ = 67°: as before, it also satisfies the equation.

Third, θ = 125°: let's check.

sin(125°) = sin(55°) ≈ 0.8192, so it would be the same as θ = 55°, but since sin(θ) is the same, it would give the same result. Similarly, θ = 113° would be sin(113°) = sin(67°), which would be the same as θ = 67°.

Now, we need to determine which of these solutions is valid based on the given information.

If θ = 55°, let's check the angle between b and a - b. Let me use the law of sines with θ = 55°.

Let |b| = 1, then |a| = sin(84°)/sin(55°) ≈ 0.9945 / 0.8192 ≈ 1.214

|a - b| = sin(29°)/sin(55°) ≈ 0.4848 / 0.8192 ≈ 0.592

Now, compute the angle between b and a - b using the dot product:

b · (a - b) = |a||b|cos(29°) - |b|² = 1.214 * 1 * 0.8746 - 1 ≈ 1.061 - 1 = 0.061

The magnitude of a - b is 0.592, so cos(angle) = 0.061 / (1 * 0.592) ≈ 0.103, which corresponds to an angle of arccos(0.103) ≈ 84°, which matches the given information! This is the correct solution.

Similarly, if θ = 67°, as before, the angle between b and a - b was 96°, which does not match the given 84°. Therefore, the correct solution is θ = 55°.

Wait, but earlier when I solved the quadratic equation, I got θ ≈ 56°, which is close to 55°. This suggests that the exact value is 55°, and the small discrepancy is due to rounding.

Let me verify with θ = 55°:

From the law of sines, the angles of the triangle would be:

- Angle opposite |a|: 84° (between b and a - b)

- Angle opposite |b|: 55° (between a and a - b)

- Angle opposite |a - b|: 29° (between a and b)

Sum: 84 + 55 + 29 = 168°, which is not 180°. Wait, this is a contradiction! What is going on?

Wait, no! The angle opposite |a - b| is the angle between a and b, which is 29°, and the other two angles are 84° and 55°, which sum to 139°, not 180°. This suggests a mistake in the assumption about the angles in the triangle.

Wait, this indicates that the law of sines approach is not directly giving the angles of the triangle, but rather the angles between the vectors. This is the confusion I had earlier. The angle between a and b is 29°, which is the angle between vectors a and b, but in the triangle, the angle opposite |a - b| is the angle between a and b, which is 29°. The angle opposite |a| is the angle between b and a - b, which is 84°, and the angle opposite |b| is the angle between a and a - b, which is θ = 55°. Therefore, the sum of angles in the triangle is 29° + 84° + 55° = 168°, which is not 180°. This is impossible, so there must be a mistake.

Wait, this suggests that the law of sines is not being applied correctly. Let me recall that in the triangle, the angles are the angles between the sides, not the angles between the vectors. The angle between vectors a and b is 29°, which is the angle between the sides |a| and |b| in the triangle. The angle between vectors b and a - b is 84°, which is the angle between the sides |b| and |a - b| in the triangle. The angle between vectors a and a - b is θ, which is the angle between the sides |a| and |a - b| in the triangle. Therefore, the sum of these three angles should be 180°, so 29° + 84° + θ = 180°, giving θ = 67°. But this contradicts the previous result of θ = 55°. What is the resolution?

The resolution is that the angle between vectors b and a - b is not the same as the angle between sides |b| and |a - b| in the triangle. This is because the vector a - b is not the side of the triangle, but the vector from the tip of b to the tip of a. Therefore, the angle between b and a - b is not the internal angle of the triangle, but rather an external angle or something else. This is why the law of sines approach gave a contradiction.

Therefore, the correct approach is the one using the quadratic equation, which gave θ ≈ 55°, and this satisfies the given condition that the angle between b and a - b is 84°. The earlier triangle angle sum approach was incorrect because it assumed the angles between vectors are the internal angles of the triangle, which is not the case.

Therefore, the correct angle is θ = 55°, but let me check with the exact calculation.

From the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0, we have two solutions: θ = 55° and θ = 67°. We determined that θ = 55° is the correct solution, as it satisfies the given angle between b and a - b being 84°. However, 55° is not one of the standard angles, but let me check if it's 55° or 56°.

Wait, when I calculated with θ = 55°, the angle between b and a - b was 84°, which matches the problem statement. Therefore, the correct answer is 55°. But let me check with θ = 55°, what is the exact value?

Wait, let me calculate the exact value using the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0. The solutions are θ = 55° and θ = 67°. But why does θ = 55° work?

Let me recall that when θ = 55°, the angle between b and a - b is 84°, which is given. When θ = 67°, the angle between b and a - b is 96°, which is not given. Therefore, θ = 55° is the correct solution.

But why does the triangle angle sum give 29 + 84 + 55 = 168°? Because the angle between vectors a and b is 29°, which is the angle between sides |a| and |b| in the triangle, the angle between vectors b and a - b is 84°, which is not the internal angle of the triangle, but the angle between vector b and vector a - b, which is different from the internal angle of the triangle. Therefore, the triangle angle sum is not directly applicable here.

Therefore, the correct answer is θ = 55°. But let me check with the exact calculation.

Wait, when I used θ = 55°, the angle between b and a - b was 84°, which matches the problem statement. Therefore, the correct answer is 55°. But let me check if 55° is the exact value or if it's a rounded value.

Let me recall that in the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0, the solutions are θ = 55° and θ = 67°. Since θ = 55° satisfies the given condition, it is the correct solution. Therefore, the angle between a and a - b is 55°.

But let me check with the initial problem statement. The angle between a and b is 29°, and the angle between b and a - b is 84°. If the angle between a and a - b is 55°, then the three angles in the triangle would be 29°, 84°, and 55°, but this sums to 168°, which is not 180°. This suggests a contradiction, but it is resolved by understanding that the angle between b and a - b is not the internal angle of the triangle.

Therefore, the correct answer is 55°. However, let me check with the exact calculation using the law of sines and the given angles.

Let me use θ = 55°, and calculate the angle between a and a - b as 55°, and verify the given angle between b and a - b is 84°.

As done earlier, with |b| = 1, |a| ≈ 1.214, |a - b| ≈ 0.592, the dot product between b and a - b is 0.061, and the angle is arccos(0.061 / (1 * 0.592)) ≈ arccos(0.103) ≈ 84°, which matches. Therefore, θ = 55° is correct.

But why does the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0 give θ = 55° as a solution? Because it is derived from the original equation, and it satisfies the given conditions.

Therefore, the correct answer is 55°. But let me check if 55° is the exact value or if it's a rounded value. Let me calculate the exact value.

Let me recall that in the equation, we had:

[sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0

This suggests that θ = 55° or θ = 67°. Since θ = 55° is the correct solution, the answer is 55°. However, let me check if 55° is the exact value or if it's an approximation.

Let me calculate the exact value using the given angles. Let me recall that 29° + 84° = 113°, and 55° + 67° = 122°, but not sure.

Alternatively, let me consider that 55° + 29° = 84°, which is the given angle between b and a - b. This suggests a relationship: the angle between a and a - b (55°) plus the angle between a and b (29°) equals the angle between b and a - b (84°). This is a possible relationship, but it's not a standard identity.

However, this suggests that the angle between a and a - b is 55°, which is 84° - 29°. Let me verify this:

If θ = 84° - 29° = 55°, then this would be the case. This seems to be a possible relationship, but I need to confirm.

Let me consider the triangle formed by vectors a, b, and a - b. If the angle between a and b is 29°, and the angle between b and a - b is 84°, then the angle between a and a - b is 180° - 29° - 84° = 67°, but this is incorrect as shown earlier. However, if the angle between a and a - b is 55°, then the angle between a and b is 29°, and the angle between b and a - b is 84°, and these three angles do not form a triangle, but the vector relationships are satisfied.

Given that the calculation with θ = 55° satisfies the given conditions, and the calculation with θ = 67° does not, the correct answer is 55°. However, let me check if 55° is the exact value or if it's an approximation.

Let me recall that in the equation [sin(θ) - sin(55°)] [sin(67°) - sin(θ)] = 0, the solutions are θ = 55° and θ = 67°. Since θ = 55° is the correct solution, and the problem likely expects an exact value, it's possible that 55° is the exact answer. However, let me check if 55° is the exact value.

Let me recall that 55° + 29° = 84°, which is the given angle between b and a - b. This suggests a possible relationship, but it's not a standard identity. However, given the calculations, it seems that 55° is the correct answer.

But let me check with the initial problem statement. The angle between a and b is 29°, and the angle between b and a - b is 84°. If I draw this, the angle between a and a - b is 55°, which is 84° - 29°. This seems to be a possible relationship, but I need to confirm.

Alternatively, let me use the law of sines with θ = 55° and see if it satisfies the original equation.

From the law of sines:

|a| / sin(84°) = |b| / sin(55°) = |a - b| / sin(29°)

Let me set |b| = 1, then |a| = sin(84°)/sin(55°) ≈ 0.9945 / 0.8192 ≈ 1.214

|a - b| = sin(29°)/sin(55°) ≈ 0.4848 / 0.8192 ≈ 0.592

Now, let's compute the angle between a and a - b using the dot product:

a · (a - b) = |a|² - a · b = |a|² - |a||b|cos(29°) ≈ (1.214)^2 - 1.214 * 1 * 0.8746 ≈ 1.474 - 1.061 ≈ 0.413

The magnitude of a is 1.214, and the magnitude of a - b is 0.592, so:

cosθ = 0.413 / (1.214 * 0.592) ≈ 0.413 / 0.719 ≈ 0.574

Which corresponds to θ ≈ 55°, since cos(55°) ≈ 0.5736. This is very close, confirming that θ = 55° is the correct solution.

Therefore, the angle between a and a - b is 55 degrees.

But wait, the problem says "the angle between b and a - b is 84°", and with θ = 55°, this is satisfied. Therefore, the correct answer is 55°.

However, let me check if there is a calculation error in the earlier steps. When I first assumed the triangle angle sum, I got 67°, but that was incorrect. The correct answer is 55°, which satisfies all conditions.

But let me check with the initial coordinate method. Let me set |b| = 1, |a| ≈ 1.214, and calculate the angle between a and a - b.

Vector a is (1.214, 0), vector b is (cos(29°), sin(29°)) ≈ (0.8746, 0.4848), vector a - b is (1.214 - 0.8746, 0 - 0.4848) ≈ (0.3394, -0.4848)

The angle between a and a - b can be calculated using the dot product:

a · (a - b) = 1.214 * 0.3394 + 0 * (-0.4848) ≈ 0.413

|a| = 1.214, |a - b| ≈ 0.592

cosθ = 0.413 / (1.214 * 0.592) ≈ 0.413 / 0.719 ≈ 0.574, which is cos(55°), so θ ≈ 55°. This confirms it.

Therefore, the correct answer is 55 degrees.

But why did the initial triangle angle sum give 67°? Because it incorrectly assumed that the angles between the vectors are the internal angles of the triangle, which is not the case. The angle between b and a - b is not the internal angle of the triangle, but rather an angle between vectors, which can be different from the internal angles.

Therefore, after careful analysis and multiple methods, the correct angle between a and a - b is 55 degrees.

**Final Answer**
\boxed{55}

To find the angle between vectors $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$, we are given:

- The angle between $\mathbf{a}$ and $\mathbf{b}$ is $29^\circ$,
- The angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ$.

---

### Step 1: Understanding the Geometry

We consider the triangle formed by the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a} - \mathbf{b}$. This triangle has:

- Side $\mathbf{a}$,
- Side $\mathbf{b}$,
- Side $\mathbf{a} - \mathbf{b}$.

The angles between the vectors are not the internal angles of the triangle, but rather the angles between the vectors themselves. This is a crucial distinction.

Let’s denote the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ as $\theta$, which is what we are trying to find.

---

### Step 2: Use of the Law of Sines

We apply the **Law of Sines** to the triangle formed by the vectors. Let:

- $\alpha = 84^\circ$ (angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$),
- $\beta = \theta$ (angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$),
- $\gamma = 29^\circ$ (angle between $\mathbf{a}$ and $\mathbf{b}$).

Then, by the Law of Sines:

$$
\frac{|\mathbf{a}|}{\sin(\alpha)} = \frac{|\mathbf{b}|}{\sin(\beta)} = \frac{|\mathbf{a} - \mathbf{b}|}{\sin(\gamma)}
$$

Let’s denote $|\mathbf{a}| = A$, $|\mathbf{b}| = B$, and $|\mathbf{a} - \mathbf{b}| = C$. Then:

$$
\frac{A}{\sin(84^\circ)} = \frac{B}{\sin(\theta)} = \frac{C}{\sin(29^\circ)}
$$

From this, we can express $A$ and $C$ in terms of $B$ and $\theta$:

$$
A = B \cdot \frac{\sin(84^\circ)}{\sin(\theta)}, \quad C = B \cdot \frac{\sin(29^\circ)}{\sin(\theta)}
$$

---

### Step 3: Use of the Law of Cosines

We also use the **Law of Cosines** to relate the magnitudes:

$$
|\mathbf{a} - \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2|\mathbf{a}||\mathbf{b}|\cos(29^\circ)
$$

Substituting the expressions for $A$ and $C$:

$$
\left(B \cdot \frac{\sin(29^\circ)}{\sin(\theta)}\right)^2 = \left(B \cdot \frac{\sin(84^\circ)}{\sin(\theta)}\right)^2 + B^2 - 2 \cdot B \cdot \frac{\sin(84^\circ)}{\sin(\theta)} \cdot B \cdot \cos(29^\circ)
$$

Divide both sides by $B^2$:

$$
\left(\frac{\sin(29^\circ)}{\sin(\theta)}\right)^2 = \left(\frac{\sin(84^\circ)}{\sin(\theta)}\right)^2 + 1 - 2 \cdot \frac{\sin(84^\circ)}{\sin(\theta)} \cdot \cos(29^\circ)
$$

Multiply through by $\sin^2(\theta)$:

$$
\sin^2(29^\circ) = \sin^2(84^\circ) + \sin^2(\theta) - 2 \sin(84^\circ) \cos(29^\circ) \sin(\theta)
$$

This is a quadratic equation in $\sin(\theta)$:

$$
\sin^2(\theta) - 2 \sin(84^\circ) \cos(29^\circ) \sin(\theta) + \left(\sin^2(84^\circ) - \sin^2(29^\circ)\right) = 0
$$

---

### Step 4: Solve the Quadratic Equation

Let’s compute the coefficients numerically:

- $\sin(84^\circ) \approx 0.9945$
- $\cos(29^\circ) \approx 0.8746$
- $\sin^2(84^\circ) \approx 0.9891$
- $\sin^2(29^\circ) \approx 0.2350$

So the equation becomes:

$$
\sin^2(\theta) - 2 \cdot 0.9945 \cdot 0.8746 \cdot \sin(\theta) + (0.9891 - 0.2350) = 0
$$

$$
\sin^2(\theta) - 1.7386 \sin(\theta) + 0.7541 = 0
$$

Solving this quadratic:

$$
\sin(\theta) = \frac{1.7386 \pm \sqrt{1.7386^2 - 4 \cdot 0.7541}}{2}
$$

$$
\sin(\theta) = \frac{1.7386 \pm \sqrt{3.022 - 3.0164}}{2} = \frac{1.7386 \pm \sqrt{0.0056}}{2}
$$

$$
\sin(\theta) = \frac{1.7386 \pm 0.075}{2}
$$

$$
\sin(\theta) \approx 0.9068 \quad \text{or} \quad \sin(\theta) \approx 0.8318
$$

Now, compute the corresponding angles:

- $\sin^{-1}(0.9068) \approx 65^\circ$
- $\sin^{-1}(0.8318) \approx 56^\circ$

---

### Step 5: Determine the Correct Angle

We now test both solutions to see which one satisfies the given angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ being $84^\circ$.

Using the **dot product**:

$$
\cos(\text{angle between } \mathbf{b} \text{ and } \mathbf{a} - \mathbf{b}) = \frac{\mathbf{b} \cdot (\mathbf{a} - \mathbf{b})}{|\mathbf{b}| |\mathbf{a} - \mathbf{b}|}
$$

We compute this for both $\theta = 56^\circ$ and $\theta = 65^\circ$, and find that only $\theta = 55^\circ$ (close to 56°) gives the correct angle of $84^\circ$ between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$.

---

### Final Answer

$$
\boxed{55}
$$

Solution:
Since $\mathbf{a}, \mathbf{b},$ and $\mathbf{a-b}$ all lie in the same plane, from the diagram below, we see that the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ - 29^\circ = \boxed{55^\circ}.$

[asy]
unitsize(5 cm);

pair A, B, C, O;

O = (0,0);
A = (1,0);
B = extension(O, O + dir(29), A, A + dir(180 - 55));
C = O + A - B;

draw(O--A,red,Arrow(6));
draw(O--B,green,Arrow(6));
draw(C--A,green,Arrow(6));
draw(O--C,blue,Arrow(6));
draw(B--A,blue,Arrow(6));

label("$\mathbf{a}$", A/2, S);
label("$\mathbf{b}$", B/2, NW);
label("$\mathbf{b}$", (A + C)/2, SE);
label("$\mathbf{a} - \mathbf{b}$", C/2, SW);
label("$\mathbf{a} - \mathbf{b}$", (A + B)/2, NE);
label("$29^\circ$", (0.2,0.05));
label("$55^\circ$", (0.15,-0.05));
[/asy]