Explanation:
Okay, so I need to compute the value of sin³18° + sin²18°. Hmm, let me think about how to approach this. I remember that sometimes trigonometric identities can help simplify expressions like this. Maybe I can use some identities related to sine functions or maybe even relate it to known angles? Let me recall some identities.

First, I know that 18° is related to the pentagon angles, right? Because in a regular pentagon, each internal angle is 108°, and the central angles are 72°, which is 2*36°, and 36° is half of 72°, but maybe there's something with 18° as well. Wait, 18° is 1/10 of 180°, so maybe it's connected to the pentagon or decagon? Not sure if that helps directly, but maybe there are exact expressions for sin18°?

Yes, I think there is a formula for sin18°. Let me try to recall. I remember that sin18° is (sqrt(5)-1)/4 multiplied by 2 or something like that. Wait, let me check. Let me try to derive it.

Let me consider a regular pentagon. In a regular pentagon, the central angles are 72°, and if you draw a star inside, you get triangles with angles of 36°, 72°, etc. Maybe using the golden ratio? The golden ratio φ is (1 + sqrt(5))/2 ≈ 1.618. I think sin18° is related to φ. Let me try to derive it.

Let me consider a triangle with angles 18°, 72°, 90°? Wait, no, 18 + 72 is 90, so maybe a right triangle? Wait, but 18°, 72°, and 90° would add up to 18 + 72 + 90 = 180°, so that's a right triangle. Wait, but is there such a triangle? Let me think. If I have a right triangle with angles 18°, 72°, and 90°, then the sides would be in some ratio. Maybe I can use some trigonometric identities here.

Alternatively, maybe using the formula for sin(5θ) or something. Since 5*18° = 90°, so maybe using multiple-angle identities. Let me try that.

Let me recall that sin(5θ) can be expressed in terms of sinθ. Let me write down the formula for sin(5θ). There's a formula for sin(5θ) in terms of sinθ. Let me try to derive it.

We know that sin(5θ) = sin(2θ + 3θ) = sin2θcos3θ + cos2θsin3θ. Then, expanding sin2θ, cos3θ, cos2θ, sin3θ. Alternatively, maybe using complex numbers or De Moivre's theorem? Maybe that's too complicated. Alternatively, use multiple-angle formulas step by step.

Alternatively, I remember that sin(5θ) = 16sin^5θ - 20sin^3θ + 5sinθ. Wait, is that correct? Let me check for θ = 0°, sin(0) = 0, and the formula gives 0 - 0 + 0 = 0, which is correct. For θ = 30°, sin(150°) = 1/2. Let me compute the formula: 16*(1/2)^5 - 20*(1/2)^3 + 5*(1/2) = 16*(1/32) - 20*(1/8) + 5/2 = 0.5 - 2.5 + 2.5 = 0.5, which matches. So the formula seems correct. Therefore, sin(5θ) = 16sin^5θ - 20sin^3θ + 5sinθ.

So if I set θ = 18°, then 5θ = 90°, so sin(5θ) = sin(90°) = 1. Therefore:

1 = 16sin^5(18°) - 20sin^3(18°) + 5sin(18°)

Let me denote x = sin(18°). Then the equation becomes:

1 = 16x^5 - 20x^3 + 5x

Rearranged:

16x^5 - 20x^3 + 5x - 1 = 0

Hmm, so we have a fifth-degree equation. Solving this might be complicated, but maybe we can factor it. Let me try to factor this equation.

First, let me check if x = 1 is a root: 16(1)^5 - 20(1)^3 + 5(1) - 1 = 16 - 20 + 5 - 1 = 0. Oh, so x = 1 is a root. Therefore, (x - 1) is a factor. Let me perform polynomial division or use synthetic division.

Dividing 16x^5 - 20x^3 + 5x - 1 by (x - 1). Wait, but the polynomial is 16x^5 + 0x^4 - 20x^3 + 0x^2 + 5x - 1.

Using synthetic division with root x = 1:

Coefficients: 16, 0, -20, 0, 5, -1

Bring down 16. Multiply by 1: 16. Add to next term: 0 + 16 = 16. Multiply by 1: 16. Add to next term: -20 + 16 = -4. Multiply by 1: -4. Add to next term: 0 + (-4) = -4. Multiply by 1: -4. Add to next term: 5 + (-4) = 1. Multiply by 1: 1. Add to last term: -1 + 1 = 0. Perfect, no remainder.

So the polynomial factors as (x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0.

So, the equation is (x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0. Since x = sin(18°) is not 1, we can focus on solving 16x^4 + 16x^3 - 4x^2 - 4x + 1 = 0.

Hmm, quartic equation. Maybe we can factor this further. Let me try grouping terms.

Let me write the quartic equation:

16x^4 + 16x^3 - 4x^2 - 4x + 1 = 0

Group terms:

(16x^4 + 16x^3) + (-4x^2 - 4x) + 1 = 0

Factor:

16x^3(x + 1) -4x(x + 1) + 1 = 0

Hmm, so (x + 1)(16x^3 - 4x) + 1 = 0. Not sure if helpful. Let me check if there are rational roots. By Rational Root Theorem, possible roots are ±1, ±1/2, ±1/4, ±1/8, ±1/16.

Let me test x = 1: 16 + 16 - 4 - 4 + 1 = 25 ≠ 0.

x = -1: 16(-1)^4 + 16(-1)^3 -4(-1)^2 -4(-1) +1 = 16 -16 -4 +4 +1 = 1 ≠ 0.

x = 1/2: 16*(1/16) + 16*(1/8) -4*(1/4) -4*(1/2) +1 = 1 + 2 -1 -2 +1 = 1 ≠ 0.

x = -1/2: 16*(1/16) + 16*(-1/8) -4*(1/4) -4*(-1/2) +1 = 1 - 2 -1 + 2 +1 = 1 ≠ 0.

x = 1/4: 16*(1/256) + 16*(1/64) -4*(1/16) -4*(1/4) +1 = (1/16) + (1/4) - (1/4) -1 +1 = 1/16 ≈ 0.0625 ≠ 0.

x = -1/4: 16*(1/256) + 16*(-1/64) -4*(1/16) -4*(-1/4) +1 = 1/16 - 1/4 - 1/4 +1 +1 = 1/16 - 1/2 + 2 = 1/16 + 3/2 = 25/16 ≠ 0.

x = 1/8: Probably too small, but let me check:

16*(1/8)^4 + 16*(1/8)^3 -4*(1/8)^2 -4*(1/8) +1. This would be 16*(1/4096) + 16*(1/512) -4*(1/64) - 4*(1/8) +1. Which is 1/256 + 1/32 - 1/16 - 1/2 +1. Converting to 256 denominator:

1 + 8 - 16 - 128 + 256 = 1 + 8 = 9; 9 -16 = -7; -7 -128 = -135; -135 +256 = 121. So 121/256 ≈ 0.472 ≠ 0.

So no rational roots. Hmm. Maybe factor into quadratics?

Let me assume that the quartic factors as (ax² + bx + c)(dx² + ex + f). Let me try to find integers a,b,c,d,e,f such that:

(ax² + bx + c)(dx² + ex + f) = 16x^4 + 16x^3 -4x^2 -4x +1.

Assuming a*d = 16. Let me try a=4, d=4. Then:

(4x² + bx + c)(4x² + ex + f) = 16x^4 + (4e + 4b)x³ + (be + 4f + 4c)x² + (bf + ce)x + cf.

Compare with original coefficients:

16x^4 + 16x^3 -4x^2 -4x +1.

So:

1. Coefficient of x^4: 4*4 = 16, correct.

2. Coefficient of x^3: 4e + 4b = 16 ⇒ e + b = 4.

3. Coefficient of x^2: be + 4f + 4c = -4.

4. Coefficient of x: bf + ce = -4.

5. Constant term: cf = 1.

Since cf = 1, possible c and f are both 1 or both -1.

Let me try c = 1, f = 1.

Then equation 5 is satisfied. Now equation 4: b*1 + e*1 = b + e = -4. Wait, but from equation 2, we have b + e = 4. Contradiction. So that's not possible.

Alternatively, c = -1, f = -1. Then equation 5: (-1)(-1) = 1, correct. Then equation 4: b*(-1) + e*(-1) = -b - e = -4 ⇒ b + e = 4. Which matches equation 2: b + e = 4. So that's consistent.

So with c = -1, f = -1, and b + e = 4.

Now equation 3: be + 4f + 4c = be + 4*(-1) + 4*(-1) = be - 4 -4 = be - 8 = -4 ⇒ be = 4.

So we have:

b + e = 4,

be = 4.

So solving for b and e. Let me set variables:

Let me think of b and e as roots of t² - 4t + 4 = 0 ⇒ (t - 2)^2 = 0 ⇒ t = 2. So b = e = 2.

Therefore, the quartic factors as (4x² + 2x -1)(4x² + 2x -1) = (4x² + 2x -1)^2?

Wait, let me check:

(4x² + 2x -1)(4x² + 2x -1) = 16x^4 + 8x³ -4x² + 8x³ + 4x² -2x -4x² -2x +1. Wait, let me compute step by step:

First, multiply 4x² by each term in the second polynomial: 4x²*4x² = 16x^4, 4x²*2x = 8x³, 4x²*(-1) = -4x².

Then multiply 2x by each term: 2x*4x² = 8x³, 2x*2x = 4x², 2x*(-1) = -2x.

Then multiply -1 by each term: -1*4x² = -4x², -1*2x = -2x, -1*(-1) = 1.

Now add all terms:

16x^4 + (8x³ + 8x³) + (-4x² + 4x² -4x²) + (-2x -2x) +1

Simplify:

16x^4 + 16x³ + (-4x²) + (-4x) +1.

Which is 16x^4 +16x³ -4x² -4x +1. Exactly the quartic we had! So yes, the quartic factors as (4x² + 2x -1)^2 = 0. Therefore, the equation is (x - 1)(4x² + 2x -1)^2 = 0.

Therefore, the solutions are x = 1 and solutions of 4x² + 2x -1 = 0. Since x = sin(18°) is between 0 and 1, and not equal to 1, we take the roots of 4x² + 2x -1 = 0.

Solving 4x² + 2x -1 = 0:

x = [-2 ± sqrt(4 + 16)] / 8 = [-2 ± sqrt(20)] / 8 = [-2 ± 2*sqrt(5)] / 8 = [-1 ± sqrt(5)] / 4.

Since sin(18°) is positive, we take the positive root: [-1 + sqrt(5)] / 4. Therefore, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait:

Wait, let me compute:

[-1 + sqrt(5)] / 4 ≈ (-1 + 2.236)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct because sin(18°) ≈ 0.3090. So yes, sin(18°) = (sqrt(5) - 1)/4 multiplied by 2? Wait, no:

Wait, the solution is x = [ -1 + sqrt(5) ] / 4? Wait, sqrt(5) ≈ 2.236, so sqrt(5) - 1 ≈ 1.236, divided by 4 is ≈ 0.309, which matches sin(18°). Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, no:

Wait, [sqrt(5) - 1]/4 is approximately (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct. However, I thought earlier that sin(18°) is (sqrt(5)-1)/4 multiplied by something? Wait, maybe I confused with another formula. Anyway, according to this derivation, sin(18°) = [sqrt(5) - 1]/4? Wait, but let me check with calculator:

sin(18°) ≈ 0.3090, and [sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, yes. So that's correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but actually, let me check with exact value. Wait, I think the exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, because if I recall, sin(18°) is (sqrt(5)-1)/4 * 2? Let me check:

Wait, another way: sometimes it's written as (sqrt(5)-1)/4 * 2, which would be (sqrt(5)-1)/2 * 1/2? Wait, no. Let me check exact value.

Alternatively, I can recall that cos(36°) = (1 + sqrt(5))/4 * 2, which is (sqrt(5)+1)/4 * 2 = (sqrt(5)+1)/2 * 1/2? Wait, no. Wait, cos(36°) is (1 + sqrt(5))/4 * 2? Let me check with calculator:

cos(36°) ≈ 0.8090. (sqrt(5) + 1)/4 ≈ (2.236 + 1)/4 ≈ 3.236 / 4 ≈ 0.809, which matches. Therefore, cos(36°) = (sqrt(5) + 1)/4 * 2? Wait, no, (sqrt(5) + 1)/4 * 2 is (sqrt(5) + 1)/2 ≈ (2.236 + 1)/2 ≈ 1.618, which is φ, the golden ratio. Wait, but cos(36°) is approximately 0.809, which is φ/2 ≈ 1.618 / 2 ≈ 0.809. Therefore, cos(36°) = φ/2 = (sqrt(5) + 1)/4 * 2? Wait, φ = (1 + sqrt(5))/2 ≈ 1.618, so φ/2 = (1 + sqrt(5))/4 * 2? Wait, no:

Wait, φ = (1 + sqrt(5))/2 ≈ 1.618. Therefore, cos(36°) = φ/2 ≈ 0.809, which is correct. Therefore, cos(36°) = (1 + sqrt(5))/4 * 2? Wait, no, cos(36°) = [1 + sqrt(5)]/4 * 2? Let me compute [1 + sqrt(5)]/4 * 2 = [1 + sqrt(5)]/2, which is φ. But cos(36°) is φ/2. Therefore, cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, I think I need to be careful here.

But regardless, according to our previous calculation, sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with exact value. Wait, actually, I think the exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, because if I compute [sqrt(5) - 1]/4 ≈ 0.309, which is correct. So maybe that's the exact value. Let me check another source in my mind. Wait, I remember that sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, let me think of another way.

Alternatively, using the formula for sin(18°). Let me recall that in a regular pentagon, the diagonal over the side is the golden ratio φ. So, if we consider a 36-72-72 triangle, splitting it into smaller triangles, maybe we can find relations. But maybe I should just accept that sin(18°) = [sqrt(5) - 1]/4? Wait, but let me check with another method.

Alternatively, using the formula for sin(θ) in terms of cos(2θ). Let me think. Let me use the identity sin(θ) = cos(90° - θ). So sin(18°) = cos(72°). And I know that cos(72°) is related to the golden ratio. And as we saw earlier, cos(72°) = [sqrt(5) - 1]/4 * 2? Wait, earlier we saw that cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, no, cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, cos(36°) is equal to (1 + sqrt(5))/4 multiplied by 2? Let me compute:

If cos(36°) = (1 + sqrt(5))/4 * 2, that would be (1 + sqrt(5))/2, which is φ ≈ 1.618, which is greater than 1, which is impossible for cosine. Therefore, that can't be. Therefore, my previous conclusion that sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309 is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with exact value.

Wait, let me check with another identity. Let me recall that sin(18°) = 2 sin(9°) cos(9°), but that might not help. Alternatively, using the formula for sin(3θ). Let me try θ = 18°, then sin(54°) = 3 sin(18°) - 4 sin³(18°). Wait, is that correct? Let me recall the formula for sin(3θ):

sin(3θ) = 3 sinθ - 4 sin³θ. Yes, that's correct. So if θ = 18°, then sin(54°) = 3 sin(18°) - 4 sin³(18°). Also, sin(54°) = cos(36°), since sin(54°) = sin(90° - 36°) = cos(36°). And we know that cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, earlier we saw that cos(36°) ≈ 0.809, which is (sqrt(5) + 1)/4 * 2? Let me compute:

Let me suppose that cos(36°) = (1 + sqrt(5))/4 * 2. Wait, (1 + sqrt(5))/4 * 2 = (1 + sqrt(5))/2 ≈ (1 + 2.236)/2 ≈ 1.618, which is φ, which is greater than 1. Not possible. Therefore, that can't be. Therefore, my previous conclusion that sin(18°) = [sqrt(5) - 1]/4 is correct. Let me check with the equation:

If sin(18°) = [sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with the equation sin(54°) = 3 sin(18°) - 4 sin³(18°). Let me compute sin(54°) ≈ 0.8090. Let me compute 3x - 4x³ where x = [sqrt(5) - 1]/4.

Let me compute x = [sqrt(5) - 1]/4 ≈ 0.3090.

Compute 3x ≈ 0.927, 4x³ ≈ 4*(0.309)^3 ≈ 4*(0.029) ≈ 0.116. Therefore, 3x - 4x³ ≈ 0.927 - 0.116 ≈ 0.811, which is approximately sin(54°) ≈ 0.809. Close enough, considering rounding errors. Therefore, this confirms that sin(18°) = [sqrt(5) - 1]/4. Wait, but actually, let me compute it exactly.

Let me compute 3x - 4x³ where x = [sqrt(5) - 1]/4.

First, compute x:

x = [sqrt(5) - 1]/4.

Compute 3x:

3x = 3[sqrt(5) - 1]/4.

Compute x³:

x³ = [ (sqrt(5) - 1)/4 ]³ = [ (sqrt(5) - 1)^3 ] / 64.

Let me expand (sqrt(5) - 1)^3:

Let me recall that (a - b)^3 = a³ - 3a²b + 3ab² - b³.

So:

(sqrt(5))³ - 3*(sqrt(5))²*1 + 3*sqrt(5)*1² - 1³

= 5*sqrt(5) - 3*5*1 + 3*sqrt(5) - 1

= 5√5 - 15 + 3√5 - 1

= (5√5 + 3√5) - 16

= 8√5 - 16.

Therefore, x³ = (8√5 - 16)/64 = (8(√5 - 2))/64 = (√5 - 2)/8.

Therefore, 4x³ = 4*(√5 - 2)/8 = (√5 - 2)/2.

Therefore, 3x - 4x³ = [3(sqrt(5) - 1)/4] - [ (√5 - 2)/2 ].

Let me compute this:

Convert to common denominator of 4:

= [3(sqrt(5) - 1)/4] - [2(√5 - 2)/4]

= [3sqrt(5) - 3 - 2sqrt(5) + 4]/4

= [ (3sqrt(5) - 2sqrt(5)) + (-3 + 4) ] / 4

= [ sqrt(5) + 1 ] / 4.

Which is equal to [sqrt(5) + 1]/4 ≈ (2.236 + 1)/4 ≈ 3.236 / 4 ≈ 0.809, which is indeed sin(54°). Therefore, this confirms that our value of x = [sqrt(5) - 1]/4 is correct for sin(18°). Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but hold on, [sqrt(5) - 1]/4 is approximately 0.309, which is correct. However, I have seen in some references that sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, let me check:

Wait, if I compute [sqrt(5) - 1]/4 ≈ 0.309, which is correct. However, sometimes it's written as (sqrt(5) - 1)/4 * 2? Wait, no, that would be [sqrt(5) - 1]/2 ≈ 1.236 / 2 ≈ 0.618, which is actually sin(36°), since sin(36°) ≈ 0.5878? Wait, no, sin(36°) ≈ 0.5878, which is different. Wait, [sqrt(5) - 1]/4 ≈ 0.309 is sin(18°), and [sqrt(5) - 1]/2 ≈ 0.618 is 2*sin(18°). Wait, 2*0.309 ≈ 0.618, which is actually 2*sin(18°). But 0.618 is approximately 1/φ, since φ ≈ 1.618, so 1/φ ≈ 0.618. Therefore, sin(18°) = (sqrt(5) - 1)/4, and 2*sin(18°) = (sqrt(5) - 1)/2 ≈ 0.618, which is 1/φ. Anyway, regardless of the exact form, we have sin(18°) = [sqrt(5) - 1]/4. Wait, but let me check with another source. Wait, according to standard references, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, according to the calculation above, it's [sqrt(5) - 1]/4. Wait, but let me check with exact value.

Alternatively, let me check with the formula for sin(π/10) which is sin(18°). The exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, according to the formula here: sin(π/10) = (sqrt(5) - 1)/4 * 2? Wait, let me check:

From the formula, sin(π/10) = (sqrt(5) - 1)/4 * 2? Let me compute:

If I recall, sin(π/10) = (sqrt(5) - 1)/4 * 2 = (sqrt(5) - 1)/2 * 1/2? No, wait:

Wait, according to the formula from the equation above, we have sin(18°) = [sqrt(5) - 1]/4? But according to the calculation with the equation, that's correct. However, according to some sources, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, let me check with decimal:

[sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct. However, if I multiply by 2, it's ≈ 0.618, which is actually 2*sin(18°). But 0.618 is approximately 1/φ, which is (sqrt(5) - 1)/2 ≈ 0.618. Therefore, sin(18°) = (sqrt(5) - 1)/4? Or is it (sqrt(5) - 1)/4 multiplied by something else?

Wait, let me check with another approach. Let me use the formula for sin(18°). Let me consider a regular pentagon and the triangle formed by two radii and a side. In a regular pentagon, the central angle is 72°, and the triangle formed by two radii and a side is an isosceles triangle with vertex angle 72° and base angles 54°. If we bisect one of the base angles, we get a 36-72-72 triangle, and so on. Maybe using the golden ratio properties.

Alternatively, let me consider a 36-72-72 triangle. Let me denote the sides. Let me suppose that the triangle has sides of length 1, 1, and x. The base angles are 72°, and the vertex angle is 36°. Then, by the law of sines:

1 / sin(72°) = x / sin(36°)

Therefore, x = [sin(36°)/sin(72°)].

But sin(72°) = 2 sin(36°) cos(36°), from the double-angle formula: sin(2θ) = 2 sinθ cosθ. Therefore, sin(72°) = 2 sin(36°) cos(36°). Therefore, x = [sin(36°) / (2 sin(36°) cos(36°))] = 1/(2 cos(36°)).

Therefore, x = 1/(2 cos(36°)). But in a regular pentagon, the ratio of diagonal to side is the golden ratio φ = (1 + sqrt(5))/2. Therefore, if the side is 1, the diagonal is φ. Therefore, in this triangle, the base is x, which is the diagonal? Wait, maybe not. Wait, in this triangle, if the two equal sides are the radii of the pentagon, and the base is the side of the pentagon. Wait, maybe I need to think differently.

Alternatively, if we have a regular pentagon with side length 1, then the diagonal is φ. If we consider a triangle formed by two diagonals and a side, maybe. This is getting too vague. Let me go back.

But regardless, we have already derived that sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309, which is correct. Therefore, I can proceed with that value.

So, given that sin(18°) = (sqrt(5) - 1)/4, let me compute sin³(18°) + sin²(18°).

Let me denote x = sin(18°) = [sqrt(5) - 1]/4.

Therefore, the expression is x³ + x².

Let me compute x² and x³.

First, compute x²:

x² = [ (sqrt(5) - 1)/4 ]² = [ (sqrt(5))² - 2*sqrt(5)*1 + 1² ] / 16 = [5 - 2sqrt(5) + 1]/16 = [6 - 2sqrt(5)] / 16 = [3 - sqrt(5)] / 8.

Then compute x³:

x³ = x² * x = [ (3 - sqrt(5))/8 ] * [ (sqrt(5) - 1)/4 ].

Let me compute this product:

Multiply numerator: (3 - sqrt(5))(sqrt(5) - 1).

Let me expand this:

3*sqrt(5) - 3*1 - sqrt(5)*sqrt(5) + sqrt(5)*1

= 3sqrt(5) - 3 - 5 + sqrt(5)

= (3sqrt(5) + sqrt(5)) - 8

= 4sqrt(5) - 8.

Therefore, x³ = [4sqrt(5) - 8]/(8*4) = [4sqrt(5) - 8]/32 = [sqrt(5) - 2]/8.

Therefore, x³ + x² = [sqrt(5) - 2]/8 + [3 - sqrt(5)]/8 = [sqrt(5) - 2 + 3 - sqrt(5)] / 8 = [1]/8 = 1/8.

Wait, that's nice! So sin³(18°) + sin²(18°) = 1/8?

Wait, let me check the calculation again to make sure.

First, x² = [3 - sqrt(5)] / 8. Correct.

x³ = [ (3 - sqrt(5))(sqrt(5) - 1) ] / 32. Wait, earlier I said denominator is 8*4 = 32? Wait, original x² is [3 - sqrt(5)] / 8, and x is [sqrt(5) - 1]/4. Therefore, multiplying them gives [ (3 - sqrt(5))(sqrt(5) - 1) ] / (8*4) = [ (3 - sqrt(5))(sqrt(5) - 1) ] / 32. Then expanding numerator:

(3)(sqrt(5)) - 3(1) - sqrt(5)(sqrt(5)) + sqrt(5)(1) = 3sqrt(5) - 3 - 5 + sqrt(5) = 4sqrt(5) - 8. Therefore, x³ = (4sqrt(5) - 8)/32 = [sqrt(5) - 2]/8. Correct.

Then x³ + x² = [sqrt(5) - 2]/8 + [3 - sqrt(5)]/8 = [sqrt(5) - 2 + 3 - sqrt(5)] / 8 = (1)/8. Yes! So the result is 1/8. Wow, that's elegant. Therefore, sin³(18°) + sin²(18°) = 1/8.

But let me verify this with approximate values to make sure.

Compute sin(18°) ≈ 0.3090.

Compute sin³(18°) ≈ (0.3090)^3 ≈ 0.029.

Compute sin²(18°) ≈ (0.3090)^2 ≈ 0.0955.

Adding them together: 0.029 + 0.0955 ≈ 0.1245.

Now, 1/8 = 0.125. Which is very close to 0.1245, considering rounding errors. Therefore, this confirms that the exact value is 1/8.

Therefore, the answer is 1/8.

But let me check once again with exact expressions.

We had:

x³ + x² = 1/8. Therefore, the answer is 1/8. Therefore, boxed answer is \boxed{\dfrac{1}{8}}.

**Final Answer**
\boxed{\dfrac{1}{8}}

To compute the value of
$$
\sin^3 18^\circ + \sin^2 18^\circ,
$$
we begin by finding an exact expression for $\sin 18^\circ$. This is a known trigonometric value that can be derived using identities or geometric properties of a regular pentagon. The exact value is:

$$
\sin 18^\circ = \frac{\sqrt{5} - 1}{4}.
$$

Let us denote this value as $x = \sin 18^\circ = \frac{\sqrt{5} - 1}{4}$.

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### Step 1: Compute $x^2$

$$
x^2 = \left( \frac{\sqrt{5} - 1}{4} \right)^2 = \frac{(\sqrt{5} - 1)^2}{16} = \frac{5 - 2\sqrt{5} + 1}{16} = \frac{6 - 2\sqrt{5}}{16} = \frac{3 - \sqrt{5}}{8}.
$$

---

### Step 2: Compute $x^3$

We use the identity $x^3 = x^2 \cdot x$:

$$
x^3 = \left( \frac{3 - \sqrt{5}}{8} \right) \cdot \left( \frac{\sqrt{5} - 1}{4} \right).
$$

Multiply the numerators:

$$
(3 - \sqrt{5})(\sqrt{5} - 1) = 3\sqrt{5} - 3 - 5 + \sqrt{5} = 4\sqrt{5} - 8.
$$

So,

$$
x^3 = \frac{4\sqrt{5} - 8}{32} = \frac{\sqrt{5} - 2}{8}.
$$

---

### Step 3: Add $x^3 + x^2$

$$
x^3 + x^2 = \frac{\sqrt{5} - 2}{8} + \frac{3 - \sqrt{5}}{8} = \frac{(\sqrt{5} - 2) + (3 - \sqrt{5})}{8} = \frac{1}{8}.
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{8}}
$$

[Assistant]
Explanation:
Okay, so I need to compute the value of sin³18° + sin²18°. Hmm, let me think about how to approach this. I remember that sometimes trigonometric identities can help simplify expressions like this. Maybe I can use some identities related to sine functions or maybe even relate it to known angles? Let me recall some identities.

First, I know that 18° is related to the pentagon angles, right? Because in a regular pentagon, each internal angle is 108°, and the central angles are 72°, which is 2*36°, and 36° is half of 72°, but maybe there's something with 18° as well. Wait, 18° is 1/10 of 180°, so maybe it's connected to the pentagon or decagon? Not sure if that helps directly, but maybe there are exact expressions for sin18°?

Yes, I think there is a formula for sin18°. Let me try to recall. I remember that sin18° is (sqrt(5)-1)/4 multiplied by 2 or something like that. Wait, let me check. Let me try to derive it.

Let me consider a regular pentagon. In a regular pentagon, the central angles are 72°, and if you draw a star inside, you get triangles with angles of 36°, 72°, etc. Maybe using the golden ratio? The golden ratio φ is (1 + sqrt(5))/2 ≈ 1.618. I think sin18° is related to φ. Let me try to derive it.

Let me consider a triangle with angles 18°, 72°, 90°? Wait, no, 18 + 72 is 90, so maybe a right triangle? Wait, but 18°, 72°, and 90° would add up to 18 + 72 + 90 = 180°, so that's a right triangle. Wait, but is there such a triangle? Let me think. If I have a right triangle with angles 18°, 72°, and 90°, then the sides would be in some ratio. Maybe I can use some trigonometric identities here.

Alternatively, maybe using the formula for sin(5θ) or something. Since 5*18° = 90°, so maybe using multiple-angle identities. Let me try that.

Let me recall that sin(5θ) can be expressed in terms of sinθ. Let me write down the formula for sin(5θ). There's a formula for sin(5θ) in terms of sinθ. Let me try to derive it.

We know that sin(5θ) = sin(2θ + 3θ) = sin2θcos3θ + cos2θsin3θ. Then, expanding sin2θ, cos3θ, cos2θ, sin3θ. Alternatively, maybe using complex numbers or De Moivre's theorem? Maybe that's too complicated. Alternatively, use multiple-angle formulas step by step.

Alternatively, I remember that sin(5θ) = 16sin^5θ - 20sin^3θ + 5sinθ. Wait, is that correct? Let me check for θ = 0°, sin(0) = 0, and the formula gives 0 - 0 + 0 = 0, which is correct. For θ = 30°, sin(150°) = 1/2. Let me compute the formula: 16*(1/2)^5 - 20*(1/2)^3 + 5*(1/2) = 16*(1/32) - 20*(1/8) + 5/2 = 0.5 - 2.5 + 2.5 = 0.5, which matches. So the formula seems correct. Therefore, sin(5θ) = 16sin^5θ - 20sin^3θ + 5sinθ.

So if I set θ = 18°, then 5θ = 90°, so sin(5θ) = sin(90°) = 1. Therefore:

1 = 16sin^5(18°) - 20sin^3(18°) + 5sin(18°)

Let me denote x = sin(18°). Then the equation becomes:

1 = 16x^5 - 20x^3 + 5x

Rearranged:

16x^5 - 20x^3 + 5x - 1 = 0

Hmm, so we have a fifth-degree equation. Solving this might be complicated, but maybe we can factor it. Let me try to factor this equation.

First, let me check if x = 1 is a root: 16(1)^5 - 20(1)^3 + 5(1) - 1 = 16 - 20 + 5 - 1 = 0. Oh, so x = 1 is a root. Therefore, (x - 1) is a factor. Let me perform polynomial division or use synthetic division.

Dividing 16x^5 - 20x^3 + 5x - 1 by (x - 1). Wait, but the polynomial is 16x^5 + 0x^4 - 20x^3 + 0x^2 + 5x - 1.

Using synthetic division with root x = 1:

Coefficients: 16, 0, -20, 0, 5, -1

Bring down 16. Multiply by 1: 16. Add to next term: 0 + 16 = 16. Multiply by 1: 16. Add to next term: -20 + 16 = -4. Multiply by 1: -4. Add to next term: 0 + (-4) = -4. Multiply by 1: -4. Add to next term: 5 + (-4) = 1. Multiply by 1: 1. Add to last term: -1 + 1 = 0. Perfect, no remainder.

So the polynomial factors as (x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0.

So, the equation is (x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0. Since x = sin(18°) is not 1, we can focus on solving 16x^4 + 16x^3 - 4x^2 - 4x + 1 = 0.

Hmm, quartic equation. Maybe we can factor this further. Let me try grouping terms.

Let me write the quartic equation:

16x^4 + 16x^3 - 4x^2 - 4x + 1 = 0

Group terms:

(16x^4 + 16x^3) + (-4x^2 - 4x) + 1 = 0

Factor:

16x^3(x + 1) -4x(x + 1) + 1 = 0

Hmm, so (x + 1)(16x^3 - 4x) + 1 = 0. Not sure if helpful. Let me check if there are rational roots. By Rational Root Theorem, possible roots are ±1, ±1/2, ±1/4, ±1/8, ±1/16.

Let me test x = 1: 16 + 16 - 4 - 4 + 1 = 25 ≠ 0.

x = -1: 16(-1)^4 + 16(-1)^3 -4(-1)^2 -4(-1) +1 = 16 -16 -4 +4 +1 = 1 ≠ 0.

x = 1/2: 16*(1/16) + 16*(1/8) -4*(1/4) -4*(1/2) +1 = 1 + 2 -1 -2 +1 = 1 ≠ 0.

x = -1/2: 16*(1/16) + 16*(-1/8) -4*(1/4) -4*(-1/2) +1 = 1 - 2 -1 + 2 +1 = 1 ≠ 0.

x = 1/4: 16*(1/256) + 16*(1/64) -4*(1/16) -4*(1/4) +1 = (1/16) + (1/4) - (1/4) -1 +1 = 1/16 ≈ 0.0625 ≠ 0.

x = -1/4: 16*(1/256) + 16*(-1/64) -4*(1/16) -4*(-1/4) +1 = 1/16 - 1/4 - 1/4 +1 +1 = 1/16 - 1/2 + 2 = 1/16 + 3/2 = 25/16 ≠ 0.

x = 1/8: Probably too small, but let me check:

16*(1/8)^4 + 16*(1/8)^3 -4*(1/8)^2 -4*(1/8) +1. This would be 16*(1/4096) + 16*(1/512) -4*(1/64) - 4*(1/8) +1. Which is 1/256 + 1/32 - 1/16 - 1/2 +1. Converting to 256 denominator:

1 + 8 - 16 - 128 + 256 = 1 + 8 = 9; 9 -16 = -7; -7 -128 = -135; -135 +256 = 121. So 121/256 ≈ 0.472 ≠ 0.

So no rational roots. Hmm. Maybe factor into quadratics?

Let me assume that the quartic factors as (ax² + bx + c)(dx² + ex + f). Let me try to find integers a,b,c,d,e,f such that:

(ax² + bx + c)(dx² + ex + f) = 16x^4 + 16x^3 -4x^2 -4x +1.

Assuming a*d = 16. Let me try a=4, d=4. Then:

(4x² + bx + c)(4x² + ex + f) = 16x^4 + (4e + 4b)x³ + (be + 4f + 4c)x² + (bf + ce)x + cf.

Compare with original coefficients:

16x^4 + 16x^3 -4x^2 -4x +1.

So:

1. Coefficient of x^4: 4*4 = 16, correct.

2. Coefficient of x^3: 4e + 4b = 16 ⇒ e + b = 4.

3. Coefficient of x^2: be + 4f + 4c = -4.

4. Coefficient of x: bf + ce = -4.

5. Constant term: cf = 1.

Since cf = 1, possible c and f are both 1 or both -1.

Let me try c = 1, f = 1.

Then equation 5 is satisfied. Now equation 4: b*1 + e*1 = b + e = -4. Wait, but from equation 2, we have b + e = 4. Contradiction. So that's not possible.

Alternatively, c = -1, f = -1. Then equation 5: (-1)(-1) = 1, correct. Then equation 4: b*(-1) + e*(-1) = -b - e = -4 ⇒ b + e = 4. Which matches equation 2: b + e = 4. So that's consistent.

So with c = -1, f = -1, and b + e = 4.

Now equation 3: be + 4f + 4c = be + 4*(-1) + 4*(-1) = be - 4 -4 = be - 8 = -4 ⇒ be = 4.

So we have:

b + e = 4,

be = 4.

So solving for b and e. Let me set variables:

Let me think of b and e as roots of t² - 4t + 4 = 0 ⇒ (t - 2)^2 = 0 ⇒ t = 2. So b = e = 2.

Therefore, the quartic factors as (4x² + 2x -1)(4x² + 2x -1) = (4x² + 2x -1)^2?

Wait, let me check:

(4x² + 2x -1)(4x² + 2x -1) = 16x^4 + 8x³ -4x² + 8x³ + 4x² -2x -4x² -2x +1. Wait, let me compute step by step:

First, multiply 4x² by each term in the second polynomial: 4x²*4x² = 16x^4, 4x²*2x = 8x³, 4x²*(-1) = -4x².

Then multiply 2x by each term: 2x*4x² = 8x³, 2x*2x = 4x², 2x*(-1) = -2x.

Then multiply -1 by each term: -1*4x² = -4x², -1*2x = -2x, -1*(-1) = 1.

Now add all terms:

16x^4 + (8x³ + 8x³) + (-4x² + 4x² -4x²) + (-2x -2x) +1

Simplify:

16x^4 + 16x³ + (-4x²) + (-4x) +1.

Which is 16x^4 +16x³ -4x² -4x +1. Exactly the quartic we had! So yes, the quartic factors as (4x² + 2x -1)^2 = 0. Therefore, the equation is (x - 1)(4x² + 2x -1)^2 = 0.

Therefore, the solutions are x = 1 and solutions of 4x² + 2x -1 = 0. Since x = sin(18°) is between 0 and 1, and not equal to 1, we take the roots of 4x² + 2x -1 = 0.

Solving 4x² + 2x -1 = 0:

x = [-2 ± sqrt(4 + 16)] / 8 = [-2 ± sqrt(20)] / 8 = [-2 ± 2*sqrt(5)] / 8 = [-1 ± sqrt(5)] / 4.

Since sin(18°) is positive, we take the positive root: [-1 + sqrt(5)] / 4. Therefore, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait:

Wait, let me compute:

[-1 + sqrt(5)] / 4 ≈ (-1 + 2.236)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct because sin(18°) ≈ 0.3090. So yes, sin(18°) = (sqrt(5) - 1)/4 multiplied by 2? Wait, no:

Wait, the solution is x = [ -1 + sqrt(5) ] / 4? Wait, sqrt(5) ≈ 2.236, so sqrt(5) - 1 ≈ 1.236, divided by 4 is ≈ 0.309, which matches sin(18°). Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, no:

Wait, [sqrt(5) - 1]/4 is approximately (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct. However, I thought earlier that sin(18°) is (sqrt(5)-1)/4 multiplied by something? Wait, maybe I confused with another formula. Anyway, according to this derivation, sin(18°) = [sqrt(5) - 1]/4? Wait, but let me check with calculator:

sin(18°) ≈ 0.3090, and [sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, yes. So that's correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but actually, let me check with exact value. Wait, I think the exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, because if I recall, sin(18°) is (sqrt(5)-1)/4 * 2? Let me check:

Wait, another way: sometimes it's written as (sqrt(5)-1)/4 * 2, which would be (sqrt(5)-1)/2 * 1/2? Wait, no. Let me check exact value.

Alternatively, I can recall that cos(36°) = (1 + sqrt(5))/4 * 2, which is (sqrt(5)+1)/4 * 2 = (sqrt(5)+1)/2 * 1/2? Wait, no. Wait, cos(36°) is (1 + sqrt(5))/4 * 2? Let me check with calculator:

cos(36°) ≈ 0.8090. (sqrt(5) + 1)/4 ≈ (2.236 + 1)/4 ≈ 3.236 / 4 ≈ 0.809, which matches. Therefore, cos(36°) = (sqrt(5) + 1)/4 * 2? Wait, no, (sqrt(5) + 1)/4 * 2 is (sqrt(5) + 1)/2 ≈ (2.236 + 1)/2 ≈ 1.618, which is φ, the golden ratio. Wait, but cos(36°) is approximately 0.809, which is φ/2 ≈ 1.618 / 2 ≈ 0.809. Therefore, cos(36°) = φ/2 = (sqrt(5) + 1)/4 * 2? Wait, φ = (1 + sqrt(5))/2 ≈ 1.618, so φ/2 = (1 + sqrt(5))/4 * 2? Wait, no:

Wait, φ = (1 + sqrt(5))/2 ≈ 1.618. Therefore, cos(36°) = φ/2 ≈ 0.809, which is correct. Therefore, cos(36°) = (1 + sqrt(5))/4 * 2? Wait, no, cos(36°) = [1 + sqrt(5)]/4 * 2? Let me compute [1 + sqrt(5)]/4 * 2 = [1 + sqrt(5)]/2, which is φ. But cos(36°) is φ/2. Therefore, cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, I think I need to be careful here.

But regardless, according to our previous calculation, sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with exact value. Wait, actually, I think the exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, because if I compute [sqrt(5) - 1]/4 ≈ 0.309, which is correct. So maybe that's the exact value. Let me check another source in my mind. Wait, I remember that sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, let me think of another way.

Alternatively, using the formula for sin(18°). Let me recall that in a regular pentagon, the diagonal over the side is the golden ratio φ. So, if we consider a 36-72-72 triangle, splitting it into smaller triangles, maybe we can find relations. But maybe I should just accept that sin(18°) = [sqrt(5) - 1]/4? Wait, but let me check with another method.

Alternatively, using the formula for sin(θ) in terms of cos(2θ). Let me think. Let me use the identity sin(θ) = cos(90° - θ). So sin(18°) = cos(72°). And I know that cos(72°) is related to the golden ratio. And as we saw earlier, cos(72°) = [sqrt(5) - 1]/4 * 2? Wait, earlier we saw that cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, no, cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, cos(36°) is equal to (1 + sqrt(5))/4 multiplied by 2? Let me compute:

If cos(36°) = (1 + sqrt(5))/4 * 2, that would be (1 + sqrt(5))/2, which is φ ≈ 1.618, which is greater than 1, which is impossible for cosine. Therefore, that can't be. Therefore, my previous conclusion that sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309 is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with exact value.

Wait, let me check with another identity. Let me recall that sin(18°) = 2 sin(9°) cos(9°), but that might not help. Alternatively, using the formula for sin(3θ). Let me try θ = 18°, then sin(54°) = 3 sin(18°) - 4 sin³(18°). Wait, is that correct? Let me recall the formula for sin(3θ):

sin(3θ) = 3 sinθ - 4 sin³θ. Yes, that's correct. So if θ = 18°, then sin(54°) = 3 sin(18°) - 4 sin³(18°). Also, sin(54°) = cos(36°), since sin(54°) = sin(90° - 36°) = cos(36°). And we know that cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, earlier we saw that cos(36°) ≈ 0.809, which is (sqrt(5) + 1)/4 * 2? Let me compute:

Let me suppose that cos(36°) = (1 + sqrt(5))/4 * 2. Wait, (1 + sqrt(5))/4 * 2 = (1 + sqrt(5))/2 ≈ (1 + 2.236)/2 ≈ 1.618, which is φ, which is greater than 1. Not possible. Therefore, that can't be. Therefore, my previous conclusion that sin(18°) = [sqrt(5) - 1]/4 is correct. Let me check with the equation:

If sin(18°) = [sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with the equation sin(54°) = 3 sin(18°) - 4 sin³(18°). Let me compute sin(54°) ≈ 0.8090. Let me compute 3x - 4x³ where x = [sqrt(5) - 1]/4.

Let me compute x = [sqrt(5) - 1]/4 ≈ 0.3090.

Compute 3x ≈ 0.927, 4x³ ≈ 4*(0.309)^3 ≈ 4*(0.029) ≈ 0.116. Therefore, 3x - 4x³ ≈ 0.927 - 0.116 ≈ 0.811, which is approximately sin(54°) ≈ 0.809. Close enough, considering rounding errors. Therefore, this confirms that sin(18°) = [sqrt(5) - 1]/4. Wait, but actually, let me compute it exactly.

Let me compute 3x - 4x³ where x = [sqrt(5) - 1]/4.

First, compute x:

x = [sqrt(5) - 1]/4.

Compute 3x:

3x = 3[sqrt(5) - 1]/4.

Compute x³:

x³ = [ (sqrt(5) - 1)/4 ]³ = [ (sqrt(5) - 1)^3 ] / 64.

Let me expand (sqrt(5) - 1)^3:

Let me recall that (a - b)^3 = a³ - 3a²b + 3ab² - b³.

So:

(sqrt(5))³ - 3*(sqrt(5))²*1 + 3*sqrt(5)*1² - 1³

= 5*sqrt(5) - 3*5*1 + 3*sqrt(5) - 1

= 5√5 - 15 + 3√5 - 1

= (5√5 + 3√5) - 16

= 8√5 - 16.

Therefore, x³ = (8√5 - 16)/64 = (8(√5 - 2))/64 = (√5 - 2)/8.

Therefore, 4x³ = 4*(√5 - 2)/8 = (√5 - 2)/2.

Therefore, 3x - 4x³ = [3(sqrt(5) - 1)/4] - [ (√5 - 2)/2 ].

Let me compute this:

Convert to common denominator of 4:

= [3(sqrt(5) - 1)/4] - [2(√5 - 2)/4]

= [3sqrt(5) - 3 - 2sqrt(5) + 4]/4

= [ (3sqrt(5) - 2sqrt(5)) + (-3 + 4) ] / 4

= [ sqrt(5) + 1 ] / 4.

Which is equal to [sqrt(5) + 1]/4 ≈ (2.236 + 1)/4 ≈ 3.236 / 4 ≈ 0.809, which is indeed sin(54°). Therefore, this confirms that our value of x = [sqrt(5) - 1]/4 is correct for sin(18°). Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but hold on, [sqrt(5) - 1]/4 is approximately 0.309, which is correct. However, I have seen in some references that sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, let me check:

Wait, if I compute [sqrt(5) - 1]/4 ≈ 0.309, which is correct. However, sometimes it's written as (sqrt(5) - 1)/4 * 2? Wait, no, that would be [sqrt(5) - 1]/2 ≈ 1.236 / 2 ≈ 0.618, which is actually sin(36°), since sin(36°) ≈ 0.5878? Wait, no, sin(36°) ≈ 0.5878, which is different. Wait, [sqrt(5) - 1]/4 ≈ 0.309 is sin(18°), and [sqrt(5) - 1]/2 ≈ 0.618 is 2*sin(18°). Wait, 2*0.309 ≈ 0.618, which is actually 2*sin(18°). But 0.618 is approximately 1/φ, since φ ≈ 1.618, so 1/φ ≈ 0.618. Therefore, sin(18°) = (sqrt(5) - 1)/4, and 2*sin(18°) = (sqrt(5) - 1)/2 ≈ 0.618, which is 1/φ. Anyway, regardless of the exact form, we have sin(18°) = [sqrt(5) - 1]/4. Wait, but let me check with another source. Wait, according to standard references, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, according to the calculation above, it's [sqrt(5) - 1]/4. Wait, but let me check with exact value.

Alternatively, let me check with the formula for sin(π/10) which is sin(18°). The exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, according to the formula here: sin(π/10) = (sqrt(5) - 1)/4 * 2? Wait, let me check:

From the formula, sin(π/10) = (sqrt(5) - 1)/4 * 2? Let me compute:

If I recall, sin(π/10) = (sqrt(5) - 1)/4 * 2 = (sqrt(5) - 1)/2 * 1/2? No, wait:

Wait, according to the formula from the equation above, we have sin(18°) = [sqrt(5) - 1]/4? But according to the calculation with the equation, that's correct. However, according to some sources, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, let me check with decimal:

[sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236 / 4 ≈ 0.309, which is correct. However, if I multiply by 2, it's ≈ 0.618, which is actually 2*sin(18°). But 0.618 is approximately 1/φ, which is (sqrt(5) - 1)/2 ≈ 0.618. Therefore, sin(18°) = (sqrt(5) - 1)/4? Or is it (sqrt(5) - 1)/4 multiplied by something else?

Wait, let me check with another approach. Let me use the formula for sin(18°). Let me consider a regular pentagon and the triangle formed by two radii and a side. In a regular pentagon, the central angle is 72°, and the triangle formed by two radii and a side is an isosceles triangle with vertex angle 72° and base angles 54°. If we bisect one of the base angles, we get a 36-72-72 triangle, and so on. Maybe using the golden ratio properties.

Alternatively, let me consider a 36-72-72 triangle. Let me denote the sides. Let me suppose that the triangle has sides of length 1, 1, and x. The base angles are 72°, and the vertex angle is 36°. Then, by the law of sines:

1 / sin(72°) = x / sin(36°)

Therefore, x = [sin(36°)/sin(72°)].

But sin(72°) = 2 sin(36°) cos(36°), from the double-angle formula: sin(2θ) = 2 sinθ cosθ. Therefore, sin(72°) = 2 sin(36°) cos(36°). Therefore, x = [sin(36°) / (2 sin(36°) cos(36°))] = 1/(2 cos(36°)).

Therefore, x = 1/(2 cos(36°)). But in a regular pentagon, the ratio of diagonal to side is the golden ratio φ = (1 + sqrt(5))/2. Therefore, if the side is 1, the diagonal is φ. Therefore, in this triangle, the base is x, which is the diagonal? Wait, maybe not. Wait, in this triangle, if the two equal sides are the radii of the pentagon, and the base is the side of the pentagon. Wait, maybe I need to think differently.

Alternatively, if we have a regular pentagon with side length 1, then the diagonal is φ. If we consider a triangle formed by two diagonals and a side, maybe. This is getting too vague. Let me go back.

But regardless, we have already derived that sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309, which is correct. Therefore, I can proceed with that value.

So, given that sin(18°) = (sqrt(5) - 1)/4, let me compute sin³(18°) + sin²(18°).

Let me denote x = sin(18°) = [sqrt(5) - 1]/4.

Therefore, the expression is x³ + x².

Let me compute x² and x³.

First, compute x²:

x² = [ (sqrt(5) - 1)/4 ]² = [ (sqrt(5))² - 2*sqrt(5)*1 + 1² ] / 16 = [5 - 2sqrt(5) + 1]/16 = [6 - 2sqrt(5)] / 16 = [3 - sqrt(5)] / 8.

Then compute x³:

x³ = x² * x = [ (3 - sqrt(5))/8 ] * [ (sqrt(5) - 1)/4 ].

Let me compute this product:

Multiply numerator: (3 - sqrt(5))(sqrt(5) - 1).

Let me expand this:

3*sqrt(5) - 3*1 - sqrt(5)*sqrt(5) + sqrt(5)*1

= 3sqrt(5) - 3 - 5 + sqrt(5)

= (3sqrt(5) + sqrt(5)) - 8

= 4sqrt(5) - 8.

Therefore, x³ = [4sqrt(5) - 8]/(8*4) = [4sqrt(5) - 8]/32 = [sqrt(5) - 2]/8.

Therefore, x³ + x² = [sqrt(5) - 2]/8 + [3 - sqrt(5)]/8 = [sqrt(5) - 2 + 3 - sqrt(5)] / 8 = [1]/8 = 1/8.

Wait, that's nice! So sin³(18°) + sin²(18°) = 1/8?

Wait, let me check the calculation again to make sure.

First, x² = [3 - sqrt(5)] / 8. Correct.

x³ = [ (3 - sqrt(5))(sqrt(5) - 1) ] / 32. Wait, earlier I said denominator is 8*4 = 32? Wait, original x² is [3 - sqrt(5)] / 8, and x is [sqrt(5) - 1]/4. Therefore, multiplying them gives [ (3 - sqrt(5))(sqrt(5) - 1) ] / (8*4) = [ (3 - sqrt(5))(sqrt(5) - 1) ] / 32. Then expanding numerator:

(3)(sqrt(5)) - 3(1) - sqrt(5)(sqrt(5)) + sqrt(5)(1) = 3sqrt(5) - 3 - 5 + sqrt(5) = 4sqrt(5) - 8. Therefore, x³ = (4sqrt(5) - 8)/32 = [sqrt(5) - 2]/8. Correct.

Then x³ + x² = [sqrt(5) - 2]/8 + [3 - sqrt(5)]/8 = [sqrt(5) - 2 + 3 - sqrt(5)] / 8 = (1)/8. Yes! So the result is 1/8. Wow, that's elegant. Therefore, sin³(18°) + sin²(18°) = 1/8.

But let me verify this with approximate values to make sure.

Compute sin(18°) ≈ 0.3090.

Compute sin³(18°) ≈ (0.3090)^3 ≈ 0.029.

Compute sin²(18°) ≈ (0.3090)^2 ≈ 0.0955.

Adding them together: 0.029 + 0.0955 ≈ 0.1245.

Now, 1/8 = 0.125. Which is very close to 0.1245, considering rounding errors. Therefore, this confirms that the exact value is 1/8.

Therefore, the answer is 1/8.

But let me check once again with exact expressions.

We had:

x³ + x² = 1/8. Therefore, the answer is 1/8. Therefore, boxed answer is \boxed{\dfrac{1}{8}}.

**Final Answer**
\boxed{\dfrac{1}{8}}

To compute the value of
$$
\sin^3 18^\circ + \sin^2 18^\circ,
$$
we begin by finding an exact expression for $\sin 18^\circ$. This is a known trigonometric value that can be derived using identities or geometric properties of a regular pentagon. The exact value is:

$$
\sin 18^\circ = \frac{\sqrt{5} - 1}{4}.
$$

Let us denote this value as $x = \sin 18^\circ = \frac{\sqrt{5} - 1}{4}$.

---

### Step 1: Compute $x^2$

$$
x^2 = \left( \frac{\sqrt{5} - 1}{4} \right)^2 = \frac{(\sqrt{5} - 1)^2}{16} = \frac{5 - 2\sqrt{5} + 1}{16} = \frac{6 - 2\sqrt{5}}{16} = \frac{3 - \sqrt{5}}{8}.
$$

---

### Step 2: Compute $x^3$

We use the identity $x^3 = x^2 \cdot x$:

$$
x^3 = \left( \frac{3 - \sqrt{5}}{8} \right) \cdot \left( \frac{\sqrt{5} - 1}{4} \right).
$$

Multiply the numerators:

$$
(3 - \sqrt{5})(\sqrt{5} - 1) = 3\sqrt{5} - 3 - 5 + \sqrt{5} = 4\sqrt{5} - 8.
$$

So,

$$
x^3 = \frac{4\sqrt{5} - 8}{32} = \frac{\sqrt{5} - 2}{8}.
$$

---

### Step 3: Add $x^3 + x^2$

$$
x^3 + x^2 = \frac{\sqrt{5} - 2}{8} + \frac{3 - \sqrt{5}}{8} = \frac{(\sqrt{5} - 2) + (3 - \sqrt{5})}{8} = \frac{1}{8}.
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{8}}
$$

Solution:
We can write
\begin{align*}
\sin^3 18^\circ + \sin^2 18^\circ &= \sin^2 18^\circ (\sin 18^\circ + 1) \\
&= \sin^2 18^\circ (\sin 18^\circ + \sin 90^\circ).
\end{align*}By sum-to-product,
\begin{align*}
\sin^2 18^\circ (\sin 18^\circ + \sin 90^\circ) &= \sin^2 18^\circ \cdot 2 \sin 54^\circ \cos 36^\circ \\
&= 2 \sin^2 18^\circ \cos^2 36^\circ \\
&= \frac{2 \sin^2 18^\circ \cos^2 18^\circ \cos^2 36^\circ}{\cos^2 18^\circ} \\
&= \frac{4 \sin^2 18^\circ \cos^2 18^\circ \cos^2 36^\circ}{2 \cos^2 18^\circ}.
\end{align*}Then by double angle formula,
\begin{align*}
\frac{4 \sin^2 18^\circ \cos^2 18^\circ \cos^2 36^\circ}{2 \cos^2 18^\circ} &= \frac{\sin^2 36^\circ \cos^2 36^\circ}{2 \cos^2 18^\circ} \\
&= \frac{4 \sin^2 36^\circ \cos^2 36^\circ}{8 \cos^2 18^\circ} \\
&= \frac{\sin^2 72^\circ}{8 \cos^2 18^\circ} \\
&= \boxed{\frac{1}{8}}.
\end{align*}Alternatively, we can plug in the value $\sin 18^\circ = \frac{\sqrt{5} - 1}{4}.$


Explanation:
Okay, so I need to find the projection of vector a onto vector b. The problem gives me vector b as (2, 6, 3) and tells me that the dot product of a and b is 8. Hmm, let me recall how projections work.

From what I remember, the projection of a onto b is a vector that represents how much of a is going in the direction of b. The formula for the projection of a onto b is (a · b / ||b||²) times b, right? Let me check that.

Yes, the formula is proj_b a = (a · b / ||b||²) * b. So I need to compute this. But wait, they didn't give me vector a directly. They only gave me the dot product of a and b, which is 8. So maybe I don't need the actual components of a? Let me think.

Since the projection formula involves a · b and the magnitude of b squared, and they already gave me a · b = 8, maybe I can compute ||b||² first. Let me calculate that.

Vector b is (2, 6, 3). The magnitude squared of b is 2² + 6² + 3². Let me compute that: 2 squared is 4, 6 squared is 36, and 3 squared is 9. Adding those up: 4 + 36 is 40, plus 9 is 49. So ||b||² is 49.

Okay, so then the scalar multiplier for the projection is (a · b)/||b||², which is 8 divided by 49. So that would be 8/49. Then the projection vector is that scalar multiplied by vector b.

So proj_b a = (8/49) * (2, 6, 3). Let me compute each component.

First component: 8/49 * 2 = 16/49. Second component: 8/49 * 6 = 48/49. Third component: 8/49 * 3 = 24/49.

Therefore, the projection of a onto b is the vector (16/49, 48/49, 24/49). Let me just verify if I did everything correctly.

Wait, the formula is (a · b / ||b||²) * b. Since a · b is 8, and ||b||² is 49, so 8/49 times b. Yes, that seems right.

Alternatively, sometimes the projection is also expressed as a scalar, which is the magnitude of the projection. But the question says "the projection of a onto b", and since they gave vector b, I think they want the vector projection.

Let me check if there's another way this could be interpreted. If they wanted the scalar projection, it would be (a · b)/||b||, which would be 8 divided by sqrt(49) = 7, so 8/7. But the question says "projection of a onto b", which usually refers to the vector projection. However, sometimes different sources might use different terminology.

But given that they provided vector b and asked for the projection, and since the answer is a vector, I think the vector projection is what they want. Also, since they didn't give me vector a, but only the dot product, it's impossible to compute the vector projection without knowing more about a. Wait, but hold on, actually, the formula for the vector projection only requires the dot product of a and b and the magnitude of b. Since they gave me the dot product, I can compute it.

So yes, my previous calculation is correct. The vector projection is (16/49, 48/49, 24/49). Let me just check the arithmetic again.

Calculating ||b||²: 2² + 6² + 3² = 4 + 36 + 9 = 49. Correct.

Then scalar multiplier: 8 / 49. Multiply each component of b by that scalar.

First component: 2*(8/49) = 16/49. Second: 6*(8/49) = 48/49. Third: 3*(8/49) = 24/49. Yep, that's right.

So unless I made a mistake in recalling the formula, this should be correct. Let me just recall the formula again.

The vector projection of a onto b is indeed (a · b / ||b||²) * b. Yes, that's the formula.

Alternatively, if I wanted to write it in terms of the scalar projection, which is the magnitude, it would be (a · b)/||b||, but that's different.

So since the problem says "projection of a onto b", and given that they gave me the dot product, which is necessary for the vector projection, I think my answer is correct.

Therefore, the projection of a onto b is the vector (16/49, 48/49, 24/49).

Wait, but maybe they want it in some other form? Like as a scalar multiple or something? But no, the projection is a vector.

Alternatively, maybe they want the scalar projection? Let me check the problem again.

The problem says: "Find the projection of a onto b = ... if a · b = 8."

Hmm. The term "projection" can sometimes be ambiguous. But in linear algebra, the projection of a onto b is usually the vector. However, sometimes in physics or other contexts, it might refer to the scalar.

But given that they provided vector b, and the answer is likely to be a vector, since if they wanted the scalar projection, they might have specified. Also, since they gave the dot product, which is needed for the vector projection.

Alternatively, maybe they want the magnitude? But again, the formula for the magnitude of the projection is |a · b| / ||b||, but that's not exactly. Wait, the scalar projection is (a · b)/||b||, which is 8 / 7.

But again, the problem says "projection of a onto b", which is a vector.

But since the problem didn't specify, but given that they gave vector b, and the answer is a vector, I think the answer is the vector.

Alternatively, maybe the problem is from a textbook or course where they define projection as scalar? But without more context, it's safer to assume vector projection.

But let me check if there's another way. Suppose I didn't remember the formula. Let me think.

The projection of a onto b is the vector in the direction of b such that it's the closest point on b to a. So, if I think geometrically, the projection vector is scalar times b.

Since we know that the dot product a · b = ||a|| ||b|| cos(theta), where theta is the angle between them.

But the projection length is ||a|| cos(theta), which is equal to (a · b)/||b||.

So scalar projection is (a · b)/||b||, which is 8 / 7.

But vector projection is scalar projection times the unit vector in the direction of b.

So that would be (8 / 7) * (b / ||b||).

Which is (8 / 7) * (2, 6, 3)/7, since ||b|| is 7.

So that would be (8 / 7) * (2/7, 6/7, 3/7) = (16/49, 48/49, 24/49).

Which matches my previous result.

Therefore, regardless of the approach, the vector projection is (16/49, 48/49, 24/49).

Therefore, I think that is the correct answer.

Just to recap:

Given:

- Vector b = [2, 6, 3]

- a · b = 8

Compute projection of a onto b.

Formula: proj_b a = (a · b / ||b||²) * b

Compute ||b||² = 2² + 6² + 3² = 4 + 36 + 9 = 49

Therefore, proj_b a = (8 / 49) * [2, 6, 3] = [16/49, 48/49, 24/49]

Yes, that seems consistent.

I think that's solid. I don't see any mistakes here. So I think that's the answer.

**Final Answer**
The projection of $\mathbf{a}$ onto $\mathbf{b}$ is \boxed{\begin{pmatrix} \dfrac{16}{49} \\ \dfrac{48}{49} \\ \dfrac{24}{49} \end{pmatrix}}.

To find the **projection of vector $\mathbf{a}$ onto vector $\mathbf{b}$**, we use the standard formula for the **vector projection**:

$$
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2} \right) \mathbf{b}
$$

---

### Step 1: Compute the magnitude squared of $\mathbf{b}$

Given:
$$
\mathbf{b} = \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}
$$

$$
\|\mathbf{b}\|^2 = 2^2 + 6^2 + 3^2 = 4 + 36 + 9 = 49
$$

---

### Step 2: Use the given dot product

We are told:
$$
\mathbf{a} \cdot \mathbf{b} = 8
$$

---

### Step 3: Apply the projection formula

$$
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{8}{49} \right) \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}
$$

Now compute each component:

- First component: $ \frac{8}{49} \cdot 2 = \frac{16}{49} $
- Second component: $ \frac{8}{49} \cdot 6 = \frac{48}{49} $
- Third component: $ \frac{8}{49} \cdot 3 = \frac{24}{49} $

---

### Final Answer

$$
\text{proj}_{\mathbf{b}} \mathbf{a} = \boxed{\begin{pmatrix} \dfrac{16}{49} \\ \dfrac{48}{49} \\ \dfrac{24}{49} \end{pmatrix}}
$$

[Assistant]
Explanation:
Okay, so I need to find the projection of vector a onto vector b. The problem gives me vector b as (2, 6, 3) and tells me that the dot product of a and b is 8. Hmm, let me recall how projections work.

From what I remember, the projection of a onto b is a vector that represents how much of a is going in the direction of b. The formula for the projection of a onto b is (a · b / ||b||²) times b, right? Let me check that.

Yes, the formula is proj_b a = (a · b / ||b||²) * b. So I need to compute this. But wait, they didn't give me vector a directly. They only gave me the dot product of a and b, which is 8. So maybe I don't need the actual components of a? Let me think.

Since the projection formula involves a · b and the magnitude of b squared, and they already gave me a · b = 8, maybe I can compute ||b||² first. Let me calculate that.

Vector b is (2, 6, 3). The magnitude squared of b is 2² + 6² + 3². Let me compute that: 2 squared is 4, 6 squared is 36, and 3 squared is 9. Adding those up: 4 + 36 is 40, plus 9 is 49. So ||b||² is 49.

Okay, so then the scalar multiplier for the projection is (a · b)/||b||², which is 8 divided by 49. So that would be 8/49. Then the projection vector is that scalar multiplied by vector b.

So proj_b a = (8/49) * (2, 6, 3). Let me compute each component.

First component: 8/49 * 2 = 16/49. Second component: 8/49 * 6 = 48/49. Third component: 8/49 * 3 = 24/49.

Therefore, the projection of a onto b is the vector (16/49, 48/49, 24/49). Let me just verify if I did everything correctly.

Wait, the formula is (a · b / ||b||²) * b. Since a · b is 8, and ||b||² is 49, so 8/49 times b. Yes, that seems right.

Alternatively, sometimes the projection is also expressed as a scalar, which is the magnitude of the projection. But the question says "the projection of a onto b", and since they gave vector b, I think they want the vector projection.

Let me check if there's another way this could be interpreted. If they wanted the scalar projection, it would be (a · b)/||b||, which would be 8 divided by sqrt(49) = 7, so 8/7. But the question says "projection of a onto b", which usually refers to the vector projection. However, sometimes different sources might use different terminology.

But given that they provided vector b and asked for the projection, and since the answer is a vector, I think the vector projection is what they want. Also, since they didn't give me vector a, but only the dot product, it's impossible to compute the vector projection without knowing more about a. Wait, but hold on, actually, the formula for the vector projection only requires the dot product of a and b and the magnitude of b. Since they gave me the dot product, I can compute it.

So yes, my previous calculation is correct. The vector projection is (16/49, 48/49, 24/49). Let me just check the arithmetic again.

Calculating ||b||²: 2² + 6² + 3² = 4 + 36 + 9 = 49. Correct.

Then scalar multiplier: 8 / 49. Multiply each component of b by that scalar.

First component: 2*(8/49) = 16/49. Second: 6*(8/49) = 48/49. Third: 3*(8/49) = 24/49. Yep, that's right.

So unless I made a mistake in recalling the formula, this should be correct. Let me just recall the formula again.

The vector projection of a onto b is indeed (a · b / ||b||²) * b. Yes, that's the formula.

Alternatively, if I wanted to write it in terms of the scalar projection, which is the magnitude, it would be (a · b)/||b||, but that's different.

So since the problem says "projection of a onto b", and given that they gave me the dot product, which is necessary for the vector projection, I think my answer is correct.

Therefore, the projection of a onto b is the vector (16/49, 48/49, 24/49).

Wait, but maybe they want it in some other form? Like as a scalar multiple or something? But no, the projection is a vector.

Alternatively, maybe they want the scalar projection? Let me check the problem again.

The problem says: "Find the projection of a onto b = ... if a · b = 8."

Hmm. The term "projection" can sometimes be ambiguous. But in linear algebra, the projection of a onto b is usually the vector. However, sometimes in physics or other contexts, it might refer to the scalar.

But given that they provided vector b, and the answer is likely to be a vector, since if they wanted the scalar projection, they might have specified. Also, since they gave the dot product, which is needed for the vector projection.

Alternatively, maybe they want the magnitude? But again, the formula for the magnitude of the projection is |a · b| / ||b||, but that's not exactly. Wait, the scalar projection is (a · b)/||b||, which is 8 / 7.

But again, the problem says "projection of a onto b", which is a vector.

But since the problem didn't specify, but given that they gave vector b, and the answer is a vector, I think the answer is the vector.

Alternatively, maybe the problem is from a textbook or course where they define projection as scalar? But without more context, it's safer to assume vector projection.

But let me check if there's another way. Suppose I didn't remember the formula. Let me think.

The projection of a onto b is the vector in the direction of b such that it's the closest point on b to a. So, if I think geometrically, the projection vector is scalar times b.

Since we know that the dot product a · b = ||a|| ||b|| cos(theta), where theta is the angle between them.

But the projection length is ||a|| cos(theta), which is equal to (a · b)/||b||.

So scalar projection is (a · b)/||b||, which is 8 / 7.

But vector projection is scalar projection times the unit vector in the direction of b.

So that would be (8 / 7) * (b / ||b||).

Which is (8 / 7) * (2, 6, 3)/7, since ||b|| is 7.

So that would be (8 / 7) * (2/7, 6/7, 3/7) = (16/49, 48/49, 24/49).

Which matches my previous result.

Therefore, regardless of the approach, the vector projection is (16/49, 48/49, 24/49).

Therefore, I think that is the correct answer.

Just to recap:

Given:

- Vector b = [2, 6, 3]

- a · b = 8

Compute projection of a onto b.

Formula: proj_b a = (a · b / ||b||²) * b

Compute ||b||² = 2² + 6² + 3² = 4 + 36 + 9 = 49

Therefore, proj_b a = (8 / 49) * [2, 6, 3] = [16/49, 48/49, 24/49]

Yes, that seems consistent.

I think that's solid. I don't see any mistakes here. So I think that's the answer.

**Final Answer**
The projection of $\mathbf{a}$ onto $\mathbf{b}$ is \boxed{\begin{pmatrix} \dfrac{16}{49} \\ \dfrac{48}{49} \\ \dfrac{24}{49} \end{pmatrix}}.

To find the **projection of vector $\mathbf{a}$ onto vector $\mathbf{b}$**, we use the standard formula for the **vector projection**:

$$
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2} \right) \mathbf{b}
$$

---

### Step 1: Compute the magnitude squared of $\mathbf{b}$

Given:
$$
\mathbf{b} = \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}
$$

$$
\|\mathbf{b}\|^2 = 2^2 + 6^2 + 3^2 = 4 + 36 + 9 = 49
$$

---

### Step 2: Use the given dot product

We are told:
$$
\mathbf{a} \cdot \mathbf{b} = 8
$$

---

### Step 3: Apply the projection formula

$$
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{8}{49} \right) \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}
$$

Now compute each component:

- First component: $ \frac{8}{49} \cdot 2 = \frac{16}{49} $
- Second component: $ \frac{8}{49} \cdot 6 = \frac{48}{49} $
- Third component: $ \frac{8}{49} \cdot 3 = \frac{24}{49} $

---

### Final Answer

$$
\text{proj}_{\mathbf{b}} \mathbf{a} = \boxed{\begin{pmatrix} \dfrac{16}{49} \\ \dfrac{48}{49} \\ \dfrac{24}{49} \end{pmatrix}}
$$

Solution:
The projection of $\mathbf{a}$ onto $\mathbf{b}$ is given by
\[\frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} = \frac{8}{2^2 + 6^2 + 3^2} \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix} = \boxed{\begin{pmatrix} 16/49 \\ 48/49 \\ 24/49 \end{pmatrix}}.\]


Explanation:
Okay, so I need to find the length of the diagonal of a square when it has the same perimeter as an equilateral triangle whose area is given as 16√3 square centimeters. Let me start by recalling some formulas related to perimeters and areas of these shapes.

First, for an equilateral triangle, all three sides are equal. Let me denote the length of each side of the triangle as 's'. The perimeter of the triangle would then be 3s. The area of an equilateral triangle is given by the formula (√3/4) * s². The problem states that the area is 16√3, so I can set up an equation:

(√3/4) * s² = 16√3

Hmm, let me solve for 's'. If I divide both sides by √3, that cancels out the √3 on both sides. So:

(1/4) * s² = 16

Multiplying both sides by 4:

s² = 64

Taking the square root of both sides:

s = 8

So each side of the equilateral triangle is 8 centimeters. Therefore, the perimeter of the triangle is 3 * 8 = 24 centimeters.

Since the square has the same perimeter, the perimeter of the square is also 24 centimeters. Let me denote the side length of the square as 'a'. The perimeter of a square is 4a, so:

4a = 24

Dividing both sides by 4:

a = 6

So each side of the square is 6 centimeters. Now, the question asks for the length of the diagonal of the square.

I remember that the diagonal of a square can be found using the Pythagorean theorem. In a square, the diagonal divides the square into two right-angled triangles, each with legs of length 'a' and hypotenuse equal to the diagonal 'd'. So:

d² = a² + a² = 2a²

Therefore, d = √(2a²) = a√2

Substituting the value of 'a' which is 6:

d = 6√2

Wait, that seems straightforward. Let me check if I did everything correctly.

First, area of the triangle: (√3/4)*s² = 16√3. Dividing both sides by √3 gives (1/4)s² = 16, so s² = 64, s = 8. That seems right. Then perimeter of triangle is 3*8=24. Square's perimeter is 24, so each side is 6. Diagonal is 6√2.

But let me think again. Maybe I made a mistake in the area formula? Let me recall. The area of an equilateral triangle is indeed (√3/4)*s². Yes, that's correct. For example, if s=2, area is √3, which is correct. So with s=8, area is (√3/4)*64 = 16√3. Perfect, that matches the given area.

So perimeter of triangle is 24, square's perimeter is 24, so square's side is 6. Diagonal is 6√2. So the answer should be 6√2.

Wait, but the problem says "Express your answer in simplest radical form." 6√2 is already in simplest radical form. So maybe that's the answer.

But let me check again. Maybe I need to verify the diagonal calculation. For a square with side length 6, the diagonal is 6√2. Yes, because if you have a square with sides of length a, the diagonal is a√2. So that's correct.

Alternatively, if I use the Pythagorean theorem: 6² + 6² = 36 + 36 = 72. Then the square root of 72 is √(36*2) = 6√2. Yep, same result.

So I think that's correct. Therefore, the diagonal of the square is 6√2 centimeters.

But let me check once more if I didn't mix up anything. The problem says the square and the triangle have equal perimeters. The area of the triangle is given, so I found the side of the triangle, then the perimeter, then the side of the square, then the diagonal. All steps seem logical.

Alternatively, maybe there's a trick here? For example, maybe the Asymptote code is showing something? Let me check the Asymptote code.

The Asymptote code draws a square from (0,0) to (1,1), then a triangle with points (1.25,0) + 1.25*dir(60), etc. But maybe that's just a generic diagram, not to scale. The problem says "a square and an equilateral triangle have equal perimeters. The area of the triangle is 16√3..." So the diagram is just illustrative.

Therefore, I think my answer is correct. The diagonal is 6√2.

But just to make sure, let me redo the steps:

1. Area of equilateral triangle: (√3/4)s² = 16√3
2. Divide both sides by √3: (1/4)s² = 16
3. Multiply by 4: s² = 64
4. Take square root: s = 8
5. Perimeter of triangle: 3*8 = 24
6. Perimeter of square: 4a = 24 => a = 6
7. Diagonal of square: a√2 = 6√2

Yes, all steps check out. Therefore, the answer is 6√2.

I think that's solid. I don't see any mistakes here. Maybe I was overcomplicating, but better safe than sorry.

**Final Answer**
The length of the diagonal of the square is \boxed{6\sqrt{2}} centimeters.

To solve the problem, we are given that a square and an equilateral triangle have **equal perimeters**, and the **area of the triangle is $16\sqrt{3}$ square centimeters**. We are asked to find the **length of the diagonal of the square**, expressed in simplest radical form.

---

### Step 1: Use the Area of the Equilateral Triangle

The formula for the area of an equilateral triangle with side length $ s $ is:

$$
\text{Area} = \frac{\sqrt{3}}{4} s^2
$$

We are told the area is $16\sqrt{3}$, so:

$$
\frac{\sqrt{3}}{4} s^2 = 16\sqrt{3}
$$

Divide both sides by $ \sqrt{3} $:

$$
\frac{1}{4} s^2 = 16
$$

Multiply both sides by 4:

$$
s^2 = 64
$$

Take the square root:

$$
s = 8
$$

So, the side length of the equilateral triangle is **8 cm**.

---

### Step 2: Find the Perimeter of the Triangle

Since the triangle is equilateral, all sides are equal. Therefore, the perimeter is:

$$
3 \times 8 = 24 \text{ cm}
$$

---

### Step 3: Use the Perimeter to Find the Side Length of the Square

The square has the same perimeter as the triangle, so its perimeter is also 24 cm. The perimeter of a square is:

$$
4a = 24
$$

Solve for $ a $:

$$
a = \frac{24}{4} = 6
$$

So, the side length of the square is **6 cm**.

---

### Step 4: Find the Diagonal of the Square

The diagonal $ d $ of a square with side length $ a $ is given by the Pythagorean theorem:

$$
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
$$

Substitute $ a = 6 $:

$$
d = 6\sqrt{2}
$$

---

### Final Answer

$$
\boxed{6\sqrt{2}}
$$

[Assistant]
Explanation:
Okay, so I need to find the length of the diagonal of a square when it has the same perimeter as an equilateral triangle whose area is given as 16√3 square centimeters. Let me start by recalling some formulas related to perimeters and areas of these shapes.

First, for an equilateral triangle, all three sides are equal. Let me denote the length of each side of the triangle as 's'. The perimeter of the triangle would then be 3s. The area of an equilateral triangle is given by the formula (√3/4) * s². The problem states that the area is 16√3, so I can set up an equation:

(√3/4) * s² = 16√3

Hmm, let me solve for 's'. If I divide both sides by √3, that cancels out the √3 on both sides. So:

(1/4) * s² = 16

Multiplying both sides by 4:

s² = 64

Taking the square root of both sides:

s = 8

So each side of the equilateral triangle is 8 centimeters. Therefore, the perimeter of the triangle is 3 * 8 = 24 centimeters.

Since the square has the same perimeter, the perimeter of the square is also 24 centimeters. Let me denote the side length of the square as 'a'. The perimeter of a square is 4a, so:

4a = 24

Dividing both sides by 4:

a = 6

So each side of the square is 6 centimeters. Now, the question asks for the length of the diagonal of the square.

I remember that the diagonal of a square can be found using the Pythagorean theorem. In a square, the diagonal divides the square into two right-angled triangles, each with legs of length 'a' and hypotenuse equal to the diagonal 'd'. So:

d² = a² + a² = 2a²

Therefore, d = √(2a²) = a√2

Substituting the value of 'a' which is 6:

d = 6√2

Wait, that seems straightforward. Let me check if I did everything correctly.

First, area of the triangle: (√3/4)*s² = 16√3. Dividing both sides by √3 gives (1/4)s² = 16, so s² = 64, s = 8. That seems right. Then perimeter of triangle is 3*8=24. Square's perimeter is 24, so each side is 6. Diagonal is 6√2.

But let me think again. Maybe I made a mistake in the area formula? Let me recall. The area of an equilateral triangle is indeed (√3/4)*s². Yes, that's correct. For example, if s=2, area is √3, which is correct. So with s=8, area is (√3/4)*64 = 16√3. Perfect, that matches the given area.

So perimeter of triangle is 24, square's perimeter is 24, so square's side is 6. Diagonal is 6√2. So the answer should be 6√2.

Wait, but the problem says "Express your answer in simplest radical form." 6√2 is already in simplest radical form. So maybe that's the answer.

But let me check again. Maybe I need to verify the diagonal calculation. For a square with side length 6, the diagonal is 6√2. Yes, because if you have a square with sides of length a, the diagonal is a√2. So that's correct.

Alternatively, if I use the Pythagorean theorem: 6² + 6² = 36 + 36 = 72. Then the square root of 72 is √(36*2) = 6√2. Yep, same result.

So I think that's correct. Therefore, the diagonal of the square is 6√2 centimeters.

But let me check once more if I didn't mix up anything. The problem says the square and the triangle have equal perimeters. The area of the triangle is given, so I found the side of the triangle, then the perimeter, then the side of the square, then the diagonal. All steps seem logical.

Alternatively, maybe there's a trick here? For example, maybe the Asymptote code is showing something? Let me check the Asymptote code.

The Asymptote code draws a square from (0,0) to (1,1), then a triangle with points (1.25,0) + 1.25*dir(60), etc. But maybe that's just a generic diagram, not to scale. The problem says "a square and an equilateral triangle have equal perimeters. The area of the triangle is 16√3..." So the diagram is just illustrative.

Therefore, I think my answer is correct. The diagonal is 6√2.

But just to make sure, let me redo the steps:

1. Area of equilateral triangle: (√3/4)s² = 16√3
2. Divide both sides by √3: (1/4)s² = 16
3. Multiply by 4: s² = 64
4. Take square root: s = 8
5. Perimeter of triangle: 3*8 = 24
6. Perimeter of square: 4a = 24 => a = 6
7. Diagonal of square: a√2 = 6√2

Yes, all steps check out. Therefore, the answer is 6√2.

I think that's solid. I don't see any mistakes here. Maybe I was overcomplicating, but better safe than sorry.

**Final Answer**
The length of the diagonal of the square is \boxed{6\sqrt{2}} centimeters.

To solve the problem, we are given that a square and an equilateral triangle have **equal perimeters**, and the **area of the triangle is $16\sqrt{3}$ square centimeters**. We are asked to find the **length of the diagonal of the square**, expressed in simplest radical form.

---

### Step 1: Use the Area of the Equilateral Triangle

The formula for the area of an equilateral triangle with side length $ s $ is:

$$
\text{Area} = \frac{\sqrt{3}}{4} s^2
$$

We are told the area is $16\sqrt{3}$, so:

$$
\frac{\sqrt{3}}{4} s^2 = 16\sqrt{3}
$$

Divide both sides by $ \sqrt{3} $:

$$
\frac{1}{4} s^2 = 16
$$

Multiply both sides by 4:

$$
s^2 = 64
$$

Take the square root:

$$
s = 8
$$

So, the side length of the equilateral triangle is **8 cm**.

---

### Step 2: Find the Perimeter of the Triangle

Since the triangle is equilateral, all sides are equal. Therefore, the perimeter is:

$$
3 \times 8 = 24 \text{ cm}
$$

---

### Step 3: Use the Perimeter to Find the Side Length of the Square

The square has the same perimeter as the triangle, so its perimeter is also 24 cm. The perimeter of a square is:

$$
4a = 24
$$

Solve for $ a $:

$$
a = \frac{24}{4} = 6
$$

So, the side length of the square is **6 cm**.

---

### Step 4: Find the Diagonal of the Square

The diagonal $ d $ of a square with side length $ a $ is given by the Pythagorean theorem:

$$
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
$$

Substitute $ a = 6 $:

$$
d = 6\sqrt{2}
$$

---

### Final Answer

$$
\boxed{6\sqrt{2}}
$$

Solution:
If we let $x = $ the side length of the triangle, then we can find the area of the triangle in terms of $x$ and then set it equal to $16 \sqrt{3}$ to find $x$. The base of the triangle has length $x$. To find the altitude, we notice that drawing an altitude splits the equilateral triangle into two $30-60-90$ triangles with the longest side having length $x$. Since the ratio of the side lengths of a $30-60-90$ triangle is $1:\sqrt{3}:2$, the altitude will have length $\frac{x\sqrt{3}}{2}$ and the area of the triangle will be $\frac{1}{2}x\left(\frac{x\sqrt{3}}{2}\right)=\frac{x^2\sqrt{3}}{4}$. Setting this equal to $16 \sqrt{3}$, we have that $\frac{x^2\sqrt{3}}{4}=16\sqrt{3}.$

Solving for $x$, we get that $x=8$. Since the side length of the triangle is $8$ and the square and triangle have equal perimeters, the square has a side length of $\frac{8 \cdot 3}{4}=6$. If we draw the diagonal of the square, we notice that it splits the square into two $45-45-90$ triangles with legs of length $6$. A $45-45-90$ triangle has side length ratios of $1:1:\sqrt{2}$, so the diagonal of the square has length $\boxed{6\sqrt{2}}$ cm.