Explanation:
Okay, so I need to figure out what happens when I rotate the complex number 3 by 90 degrees clockwise around the point -4 -5i. Hmm, rotations in the complex plane... Let me recall how that works.

First, I remember that rotating a complex number around another point involves a few steps. I think the general approach is to translate the complex plane so that the center of rotation becomes the origin, perform the rotation, and then translate back. Let me verify that.

So, if I want to rotate a point z around a center point c, the formula should be something like: rotate (z - c) by the angle, then add c back. Right? So, mathematically, it would be:

rotated_z = (z - c) * e^{-iθ} + c

Wait, because rotating clockwise by θ is equivalent to multiplying by e^{-iθ}, whereas counterclockwise would be e^{iθ}. Since we're rotating clockwise by 90 degrees, θ is 90 degrees, which is π/2 radians. So e^{-iπ/2} is equal to cos(-π/2) + i sin(-π/2) which is 0 - i. So that simplifies to -i.

Let me write that down step by step.

Given:
- Original complex number: z = 3 (which is 3 + 0i)
- Center of rotation: c = -4 -5i
- Rotation angle: 90 degrees clockwise, which corresponds to multiplying by -i

So, the formula is:

rotated_z = (z - c) * (-i) + c

Let me compute z - c first.

z - c = (3) - (-4 -5i) = 3 + 4 + 5i = 7 + 5i

Then multiply this by -i:

(7 + 5i) * (-i)

Let me compute that. Distribute the multiplication:

7*(-i) + 5i*(-i) = -7i -5i²

Since i² = -1, so -5i² = -5*(-1) = 5

Therefore, the result is 5 -7i

Then add back the center c, which is -4 -5i:

rotated_z = (5 -7i) + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i

Wait, so the resulting complex number is 1 -12i? Let me check my steps again to make sure I didn't make a mistake.

First, z - c: 3 - (-4 -5i) = 3 +4 +5i = 7 +5i. That seems right.

Multiplying by -i: (7 +5i)(-i). Let me do that again.

Multiply 7 by -i: -7i. Multiply 5i by -i: -5i². Which is -5*(-1) = 5. So total is 5 -7i. Then adding back c: -4 -5i.

So 5 -7i + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i.

Hmm, seems correct. Let me think if there's another way to approach this. Maybe using rotation matrices?

Alternatively, since complex numbers can be represented as vectors, rotating a point around another point can be done by translating the system. Let me try that approach.

Let me consider the complex plane. The original point is 3, which is (3, 0) in coordinates. The center of rotation is (-4, -5).

To rotate around (-4, -5), first translate the system so that (-4, -5) becomes the origin. So subtract (-4, -5) from the original point.

Original point translated: (3 - (-4), 0 - (-5)) = (7, 5). Which is the same as 7 +5i, which matches the earlier calculation.

Now, rotating this point 90 degrees clockwise. The rotation matrix for 90 degrees clockwise is:

[ 0, 1]
[-1, 0]

So applying that to the vector (7, 5):

x' = 0*7 + 1*5 = 5

y' = -1*7 + 0*5 = -7

So the new coordinates after rotation are (5, -7), which corresponds to 5 -7i. Then translate back by adding (-4, -5):

5 + (-4) = 1

-7 + (-5) = -12

So the final coordinates are (1, -12), which is 1 -12i.

Same result as before. So that seems to confirm that 1 -12i is the correct answer.

Wait, but let me think again. Maybe I can visualize this? Let me try to imagine the rotation.

Original point is 3 on the real axis. The center of rotation is at (-4, -5), which is in the third quadrant. Rotating 3 around that point 90 degrees clockwise...

But maybe it's hard to visualize. Alternatively, maybe I can check with another method.

Another way to think about rotation: multiplying by e^{-iθ} is equivalent to rotating the complex number. But since we are rotating around a point, not the origin, we need to adjust.

Alternatively, maybe using complex transformations.

But I think the two methods I used already confirm that the answer is 1 -12i.

Wait, but let me check once more with another approach. Let me recall that rotating a complex number z around a point c by θ degrees clockwise is equivalent to:

rotated_z = (z - c) * e^{-iθ} + c

Which is exactly what I did. So θ is 90 degrees, so e^{-iθ} is e^{-iπ/2} = -i.

So (z - c) is 7 +5i, multiplied by -i gives 5 -7i, then adding c gives 5 -7i + (-4 -5i) = 1 -12i.

Yes, same result.

Alternatively, maybe I can use vectors. Let me think.

The vector from the center to the original point is z - c = 7 +5i. Rotating this vector 90 degrees clockwise would be multiplying by -i, as before.

So the new vector is -i*(7 +5i) = -7i -5i² = -7i +5 = 5 -7i. Then adding back the center c gives 5 -7i + (-4 -5i) = 1 -12i.

Same answer again.

Therefore, I think I can be confident that the answer is 1 -12i.

But just to make sure, let me try plugging in numbers. Let me consider coordinates.

Original point: (3, 0). Center: (-4, -5).

Translate the original point by adding (4, 5) to move the center to origin: (3 +4, 0 +5) = (7, 5).

Rotate 90 degrees clockwise: (x, y) becomes (y, -x). Wait, is that right?

Yes, for a 90-degree clockwise rotation, the rotation matrix is:

[ 0, 1]
[-1, 0]

So (x, y) becomes (0*x + 1*y, -1*x + 0*y) = (y, -x).

So applying that to (7, 5): (5, -7). Then translate back by subtracting (4, 5)? Wait, no. Wait, when you translate back, you need to add the original center? Wait, no. Wait, when you translate the original point by adding (4,5), then after rotation, you need to translate back by subtracting (4,5)? Wait, no. Wait, the translation to move the center to origin is subtracting the center. So to reverse it, you need to add the center.

Wait, let me think again.

Original point: P. Center: C.

To rotate around C, you translate P by -C to get P', rotate P', then translate back by +C.

So P' = P - C. Then rotated P'' = P' rotated. Then final point is P'' + C.

So in coordinates, if original point is (3, 0), center is (-4, -5).

Translate P by -C: (3 - (-4), 0 - (-5)) = (7, 5).

Rotate 90 degrees clockwise: (5, -7).

Translate back by adding C: (5 + (-4), -7 + (-5)) = (1, -12).

Which is (1, -12), so complex number 1 -12i.

Yes, same result.

Therefore, I think this is correct.

Alternatively, maybe I can check with another angle. Suppose I rotate 3 around -4 -5i by 90 degrees clockwise. Let me think about the distance between the original point and the center.

The distance between 3 and -4 -5i is |3 - (-4 -5i)| = |7 +5i| = sqrt(7² +5²) = sqrt(49 +25) = sqrt(74).

After rotation, the distance should remain the same. Let me check the distance between 1 -12i and -4 -5i.

Compute |(1 -12i) - (-4 -5i)| = |1 +4 -12i +5i| = |5 -7i| = sqrt(25 + 49) = sqrt(74).

Same distance. So that checks out.

Also, the angle between the original vector and the new vector should be 90 degrees. Let me check the original vector from center to original point: 7 +5i. The new vector is 5 -7i.

The original vector is (7,5), new vector is (5, -7). The dot product is 7*5 + 5*(-7) = 35 -35 = 0. So they are perpendicular. Which confirms that the rotation was indeed 90 degrees.

Therefore, all checks confirm that the result is 1 -12i.

So I think that's the correct answer.

**Final Answer**
The resulting complex number is \boxed{1 - 12i}.

To determine the result of a $90^\circ$ clockwise rotation of the complex number $3$ around the point $-4 - 5i$, we follow a standard geometric transformation approach in the complex plane.

---

### Step 1: Understand the Rotation Formula

To rotate a complex number $z$ around a center $c$ by an angle $\theta$ **clockwise**, we use the formula:

$$
\text{rotated}_z = (z - c) \cdot e^{-i\theta} + c
$$

Here:
- $z = 3$ (the point to be rotated)
- $c = -4 - 5i$ (the center of rotation)
- $\theta = 90^\circ = \frac{\pi}{2}$ radians

Since the rotation is **clockwise**, we use $e^{-i\theta} = e^{-i\pi/2} = -i$

---

### Step 2: Translate the Point

First, subtract the center from the original point:

$$
z - c = 3 - (-4 - 5i) = 3 + 4 + 5i = 7 + 5i
$$

---

### Step 3: Apply the Rotation

Now, multiply the translated point by $-i$:

$$
(7 + 5i)(-i) = -7i - 5i^2 = -7i + 5 = 5 - 7i
$$

(Note: $i^2 = -1$, so $-5i^2 = 5$)

---

### Step 4: Translate Back

Add the center back to the rotated point:

$$
(5 - 7i) + (-4 - 5i) = (5 - 4) + (-7i - 5i) = 1 - 12i
$$

---

### Final Result

Thus, the resulting complex number after a $90^\circ$ clockwise rotation of $3$ around $-4 - 5i$ is:

$$
\boxed{1 - 12i}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what happens when I rotate the complex number 3 by 90 degrees clockwise around the point -4 -5i. Hmm, rotations in the complex plane... Let me recall how that works.

First, I remember that rotating a complex number around another point involves a few steps. I think the general approach is to translate the complex plane so that the center of rotation becomes the origin, perform the rotation, and then translate back. Let me verify that.

So, if I want to rotate a point z around a center point c, the formula should be something like: rotate (z - c) by the angle, then add c back. Right? So, mathematically, it would be:

rotated_z = (z - c) * e^{-iθ} + c

Wait, because rotating clockwise by θ is equivalent to multiplying by e^{-iθ}, whereas counterclockwise would be e^{iθ}. Since we're rotating clockwise by 90 degrees, θ is 90 degrees, which is π/2 radians. So e^{-iπ/2} is equal to cos(-π/2) + i sin(-π/2) which is 0 - i. So that simplifies to -i.

Let me write that down step by step.

Given:
- Original complex number: z = 3 (which is 3 + 0i)
- Center of rotation: c = -4 -5i
- Rotation angle: 90 degrees clockwise, which corresponds to multiplying by -i

So, the formula is:

rotated_z = (z - c) * (-i) + c

Let me compute z - c first.

z - c = (3) - (-4 -5i) = 3 + 4 + 5i = 7 + 5i

Then multiply this by -i:

(7 + 5i) * (-i)

Let me compute that. Distribute the multiplication:

7*(-i) + 5i*(-i) = -7i -5i²

Since i² = -1, so -5i² = -5*(-1) = 5

Therefore, the result is 5 -7i

Then add back the center c, which is -4 -5i:

rotated_z = (5 -7i) + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i

Wait, so the resulting complex number is 1 -12i? Let me check my steps again to make sure I didn't make a mistake.

First, z - c: 3 - (-4 -5i) = 3 +4 +5i = 7 +5i. That seems right.

Multiplying by -i: (7 +5i)(-i). Let me do that again.

Multiply 7 by -i: -7i. Multiply 5i by -i: -5i². Which is -5*(-1) = 5. So total is 5 -7i. Then adding back c: -4 -5i.

So 5 -7i + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i.

Hmm, seems correct. Let me think if there's another way to approach this. Maybe using rotation matrices?

Alternatively, since complex numbers can be represented as vectors, rotating a point around another point can be done by translating the system. Let me try that approach.

Let me consider the complex plane. The original point is 3, which is (3, 0) in coordinates. The center of rotation is (-4, -5).

To rotate around (-4, -5), first translate the system so that (-4, -5) becomes the origin. So subtract (-4, -5) from the original point.

Original point translated: (3 - (-4), 0 - (-5)) = (7, 5). Which is the same as 7 +5i, which matches the earlier calculation.

Now, rotating this point 90 degrees clockwise. The rotation matrix for 90 degrees clockwise is:

[ 0, 1]
[-1, 0]

So applying that to the vector (7, 5):

x' = 0*7 + 1*5 = 5

y' = -1*7 + 0*5 = -7

So the new coordinates after rotation are (5, -7), which corresponds to 5 -7i. Then translate back by adding (-4, -5):

5 + (-4) = 1

-7 + (-5) = -12

So the final coordinates are (1, -12), which is 1 -12i.

Same result as before. So that seems to confirm that 1 -12i is the correct answer.

Wait, but let me think again. Maybe I can visualize this? Let me try to imagine the rotation.

Original point is 3 on the real axis. The center of rotation is at (-4, -5), which is in the third quadrant. Rotating 3 around that point 90 degrees clockwise...

But maybe it's hard to visualize. Alternatively, maybe I can check with another method.

Another way to think about rotation: multiplying by e^{-iθ} is equivalent to rotating the complex number. But since we are rotating around a point, not the origin, we need to adjust.

Alternatively, maybe using complex transformations.

But I think the two methods I used already confirm that the answer is 1 -12i.

Wait, but let me check once more with another approach. Let me recall that rotating a complex number z around a point c by θ degrees clockwise is equivalent to:

rotated_z = (z - c) * e^{-iθ} + c

Which is exactly what I did. So θ is 90 degrees, so e^{-iθ} is e^{-iπ/2} = -i.

So (z - c) is 7 +5i, multiplied by -i gives 5 -7i, then adding c gives 5 -7i + (-4 -5i) = 1 -12i.

Yes, same result.

Alternatively, maybe I can use vectors. Let me think.

The vector from the center to the original point is z - c = 7 +5i. Rotating this vector 90 degrees clockwise would be multiplying by -i, as before.

So the new vector is -i*(7 +5i) = -7i -5i² = -7i +5 = 5 -7i. Then adding back the center c gives 5 -7i + (-4 -5i) = 1 -12i.

Same answer again.

Therefore, I think I can be confident that the answer is 1 -12i.

But just to make sure, let me try plugging in numbers. Let me consider coordinates.

Original point: (3, 0). Center: (-4, -5).

Translate the original point by adding (4, 5) to move the center to origin: (3 +4, 0 +5) = (7, 5).

Rotate 90 degrees clockwise: (x, y) becomes (y, -x). Wait, is that right?

Yes, for a 90-degree clockwise rotation, the rotation matrix is:

[ 0, 1]
[-1, 0]

So (x, y) becomes (0*x + 1*y, -1*x + 0*y) = (y, -x).

So applying that to (7, 5): (5, -7). Then translate back by subtracting (4, 5)? Wait, no. Wait, when you translate back, you need to add the original center? Wait, no. Wait, when you translate the original point by adding (4,5), then after rotation, you need to translate back by subtracting (4,5)? Wait, no. Wait, the translation to move the center to origin is subtracting the center. So to reverse it, you need to add the center.

Wait, let me think again.

Original point: P. Center: C.

To rotate around C, you translate P by -C to get P', rotate P', then translate back by +C.

So P' = P - C. Then rotated P'' = P' rotated. Then final point is P'' + C.

So in coordinates, if original point is (3, 0), center is (-4, -5).

Translate P by -C: (3 - (-4), 0 - (-5)) = (7, 5).

Rotate 90 degrees clockwise: (5, -7).

Translate back by adding C: (5 + (-4), -7 + (-5)) = (1, -12).

Which is (1, -12), so complex number 1 -12i.

Yes, same result.

Therefore, I think this is correct.

Alternatively, maybe I can check with another angle. Suppose I rotate 3 around -4 -5i by 90 degrees clockwise. Let me think about the distance between the original point and the center.

The distance between 3 and -4 -5i is |3 - (-4 -5i)| = |7 +5i| = sqrt(7² +5²) = sqrt(49 +25) = sqrt(74).

After rotation, the distance should remain the same. Let me check the distance between 1 -12i and -4 -5i.

Compute |(1 -12i) - (-4 -5i)| = |1 +4 -12i +5i| = |5 -7i| = sqrt(25 + 49) = sqrt(74).

Same distance. So that checks out.

Also, the angle between the original vector and the new vector should be 90 degrees. Let me check the original vector from center to original point: 7 +5i. The new vector is 5 -7i.

The original vector is (7,5), new vector is (5, -7). The dot product is 7*5 + 5*(-7) = 35 -35 = 0. So they are perpendicular. Which confirms that the rotation was indeed 90 degrees.

Therefore, all checks confirm that the result is 1 -12i.

So I think that's the correct answer.

**Final Answer**
The resulting complex number is \boxed{1 - 12i}.

To determine the result of a $90^\circ$ clockwise rotation of the complex number $3$ around the point $-4 - 5i$, we follow a standard geometric transformation approach in the complex plane.

---

### Step 1: Understand the Rotation Formula

To rotate a complex number $z$ around a center $c$ by an angle $\theta$ **clockwise**, we use the formula:

$$
\text{rotated}_z = (z - c) \cdot e^{-i\theta} + c
$$

Here:
- $z = 3$ (the point to be rotated)
- $c = -4 - 5i$ (the center of rotation)
- $\theta = 90^\circ = \frac{\pi}{2}$ radians

Since the rotation is **clockwise**, we use $e^{-i\theta} = e^{-i\pi/2} = -i$

---

### Step 2: Translate the Point

First, subtract the center from the original point:

$$
z - c = 3 - (-4 - 5i) = 3 + 4 + 5i = 7 + 5i
$$

---

### Step 3: Apply the Rotation

Now, multiply the translated point by $-i$:

$$
(7 + 5i)(-i) = -7i - 5i^2 = -7i + 5 = 5 - 7i
$$

(Note: $i^2 = -1$, so $-5i^2 = 5$)

---

### Step 4: Translate Back

Add the center back to the rotated point:

$$
(5 - 7i) + (-4 - 5i) = (5 - 4) + (-7i - 5i) = 1 - 12i
$$

---

### Final Result

Thus, the resulting complex number after a $90^\circ$ clockwise rotation of $3$ around $-4 - 5i$ is:

$$
\boxed{1 - 12i}
$$

Solution:
A $90^\circ$ rotation in the clockwise direction corresponds to multiplication by $\operatorname{cis} (-90^\circ) = -i.$

[asy]
unitsize(0.4 cm);

pair O = (-4,-5), A = (3,0), B = rotate(-90,O)*(A);

draw(O--A,dashed);
draw(O--B,dashed);

dot("$3$", A, NE);
dot("$1 - 12i$", B, SE);
dot("$-4 - 5i$", O, W);
[/asy]

Let $z$ be the image of $3$ under the rotation.  Since center of the rotation is $-4 - 5i,$
\[z - (-4 - 5i) = (-i)(3 - (-4 - 5i)).\]Solving, we find $z = \boxed{1 - 12i}.$


Explanation:
Okay, let's see. There's a problem about witches and sorcerers at an arcane mixer, and we need to figure out how many handshakes happened. Let me try to break this down step by step.

First, the problem says there are six witches and ten sorcerers. So, total people at the mixer would be 6 + 10 = 16 people. But I don't know if that's directly relevant yet.

The witches decided to shake hands with each of the sorcerers. So, does that mean each witch shakes hands with each sorcerer? Let me make sure I understand. The problem says "the witches have decided to shake hands with each of the sorcerers." So, maybe each witch shakes hands with every sorcerer? That would be a handshake between every witch and every sorcerer.

But then it says none of the witches shook hands with each other. So, witches don't shake hands with other witches. And the sorcerers all hate each other and didn't shake hands with other sorcerers. So, sorcerers don't shake hands with other sorcerers either.

So, the only handshakes that happened are between witches and sorcerers. So, each witch shakes hands with each sorcerer.

So, if there are 6 witches and 10 sorcerers, then the number of handshakes would be 6 times 10? Let me check that.

Wait, if each witch shakes hands with each sorcerer, then for each witch, there are 10 handshakes. So, 6 witches would have 6*10 handshakes. But is there any overcounting here? Like, if I count each handshake from the witch's perspective, does that count each handshake once?

Yes, because each handshake is between one witch and one sorcerer. So, if I count all the handshakes from the witches' side, each handshake is unique. So, 6*10 = 60 handshakes.

Alternatively, I can think of it as the number of ways to choose a witch and a sorcerer, since each handshake is between one witch and one sorcerer. So, that would be combinations where we pick one witch out of 6 and one sorcerer out of 10.

So, the formula for that would be 6C1 * 10C1, which is 6*10 = 60.

Alternatively, if I think about the total number of possible handshakes without any restrictions, it would be C(16,2) which is 16*15/2 = 120. But then we have to subtract the handshakes that didn't happen.

Wait, maybe that's another way to approach it. Let me try that.

Total possible handshakes if everyone shook hands with everyone else: C(16,2) = 120.

But then we have to subtract the handshakes that didn't happen.

The problem says witches didn't shake hands with each other. So, how many handshakes are between witches? There are 6 witches, so the number of handshakes between witches would be C(6,2) = 15.

Similarly, the sorcerers didn't shake hands with each other. There are 10 sorcerers, so the number of handshakes between sorcerers is C(10,2) = 45.

Therefore, total handshakes that didn't happen are 15 + 45 = 60.

Therefore, the total handshakes that did happen would be total possible handshakes minus the ones that didn't happen: 120 - 60 = 60.

So, that gives the same answer as before.

So, either way, whether I calculate directly the number of witch-sorcerer handshakes, or calculate total handshakes and subtract the ones that didn't happen, I get 60.

Therefore, the answer should be 60.

But let me just make sure I didn't make any mistakes here.

First approach: 6 witches each shaking hands with 10 sorcerers. So, 6*10=60.

Second approach: total handshakes C(16,2)=120. Subtract the ones between witches C(6,2)=15 and between sorcerers C(10,2)=45. 15+45=60. 120-60=60.

Same answer.

Alternatively, maybe think of it as a bipartite graph, where there are two groups: witches and sorcerers. The handshakes are the edges between the two groups. So, the number of edges is the product of the number of nodes in each group. So, 6*10=60.

Yes, that's another way to think about it.

So, all methods lead to 60.

Therefore, I think the answer is 60.

But let me check if there's any possible mistake here.

Wait, the problem says "the witches have decided to shake hands with each of the sorcerers." So, does that mean that each witch shakes hands with each sorcerer? Or does it mean that each witch shakes hands with each sorcerer once? Well, in handshake problems, usually, each handshake is between two people, and it's counted once. So, if each witch shakes hands with each sorcerer, then it's 6*10=60.

Alternatively, if there was some other restriction, but the problem says that witches don't shake hands with each other, and sorcerers don't shake hands with each other. So, all the handshakes are between witches and sorcerers.

Therefore, the answer is 60.

I think that's solid.

**Final Answer**
The number of handshakes that took place at the mixer is \boxed{60}.

To determine the number of handshakes that occurred at the arcane mixer, we begin by analyzing the structure of the group and the rules governing the handshakes.

---

### **Step 1: Understanding the Participants**

- There are **6 witches** and **10 sorcerers**, making a total of **16 individuals**.
- **Witches** only shake hands with **sorcerers**.
- **Witches do not shake hands with other witches**.
- **Sorcerers do not shake hands with other sorcerers**.

This means that **no handshakes occur within the same group** (witches with witches, or sorcerers with sorcerers), and **all handshakes occur between witches and sorcerers**.

---

### **Step 2: Counting the Handshakes**

Each of the **6 witches** shakes hands with **each of the 10 sorcerers**. Since every witch-sorcerer pair shakes hands exactly once, the total number of handshakes is:

$$
6 \text{ witches} \times 10 \text{ sorcerers} = 60 \text{ handshakes}
$$

This is a straightforward multiplication, as each handshake is uniquely between one witch and one sorcerer.

---

### **Step 3: Alternative Verification via Total Handshakes**

We can also verify this by considering the total number of possible handshakes among all 16 individuals, and then subtracting the ones that **did not occur**.

- **Total possible handshakes** (if everyone shook hands with everyone else):
  $$
  \binom{16}{2} = \frac{16 \times 15}{2} = 120
  $$

- **Handshakes that did not occur**:
  - Between witches: $\binom{6}{2} = 15$
  - Between sorcerers: $\binom{10}{2} = 45$
  - Total non-handshakes: $15 + 45 = 60$

- **Handshakes that did occur**:
  $$
  120 - 60 = 60
  $$

This confirms the earlier result.

---

### **Conclusion**

All methods lead to the same conclusion: the number of handshakes that took place at the mixer is:

$$
\boxed{60}
$$

[Assistant]
Explanation:
Okay, let's see. There's a problem about witches and sorcerers at an arcane mixer, and we need to figure out how many handshakes happened. Let me try to break this down step by step.

First, the problem says there are six witches and ten sorcerers. So, total people at the mixer would be 6 + 10 = 16 people. But I don't know if that's directly relevant yet.

The witches decided to shake hands with each of the sorcerers. So, does that mean each witch shakes hands with each sorcerer? Let me make sure I understand. The problem says "the witches have decided to shake hands with each of the sorcerers." So, maybe each witch shakes hands with every sorcerer? That would be a handshake between every witch and every sorcerer.

But then it says none of the witches shook hands with each other. So, witches don't shake hands with other witches. And the sorcerers all hate each other and didn't shake hands with other sorcerers. So, sorcerers don't shake hands with other sorcerers either.

So, the only handshakes that happened are between witches and sorcerers. So, each witch shakes hands with each sorcerer.

So, if there are 6 witches and 10 sorcerers, then the number of handshakes would be 6 times 10? Let me check that.

Wait, if each witch shakes hands with each sorcerer, then for each witch, there are 10 handshakes. So, 6 witches would have 6*10 handshakes. But is there any overcounting here? Like, if I count each handshake from the witch's perspective, does that count each handshake once?

Yes, because each handshake is between one witch and one sorcerer. So, if I count all the handshakes from the witches' side, each handshake is unique. So, 6*10 = 60 handshakes.

Alternatively, I can think of it as the number of ways to choose a witch and a sorcerer, since each handshake is between one witch and one sorcerer. So, that would be combinations where we pick one witch out of 6 and one sorcerer out of 10.

So, the formula for that would be 6C1 * 10C1, which is 6*10 = 60.

Alternatively, if I think about the total number of possible handshakes without any restrictions, it would be C(16,2) which is 16*15/2 = 120. But then we have to subtract the handshakes that didn't happen.

Wait, maybe that's another way to approach it. Let me try that.

Total possible handshakes if everyone shook hands with everyone else: C(16,2) = 120.

But then we have to subtract the handshakes that didn't happen.

The problem says witches didn't shake hands with each other. So, how many handshakes are between witches? There are 6 witches, so the number of handshakes between witches would be C(6,2) = 15.

Similarly, the sorcerers didn't shake hands with each other. There are 10 sorcerers, so the number of handshakes between sorcerers is C(10,2) = 45.

Therefore, total handshakes that didn't happen are 15 + 45 = 60.

Therefore, the total handshakes that did happen would be total possible handshakes minus the ones that didn't happen: 120 - 60 = 60.

So, that gives the same answer as before.

So, either way, whether I calculate directly the number of witch-sorcerer handshakes, or calculate total handshakes and subtract the ones that didn't happen, I get 60.

Therefore, the answer should be 60.

But let me just make sure I didn't make any mistakes here.

First approach: 6 witches each shaking hands with 10 sorcerers. So, 6*10=60.

Second approach: total handshakes C(16,2)=120. Subtract the ones between witches C(6,2)=15 and between sorcerers C(10,2)=45. 15+45=60. 120-60=60.

Same answer.

Alternatively, maybe think of it as a bipartite graph, where there are two groups: witches and sorcerers. The handshakes are the edges between the two groups. So, the number of edges is the product of the number of nodes in each group. So, 6*10=60.

Yes, that's another way to think about it.

So, all methods lead to 60.

Therefore, I think the answer is 60.

But let me check if there's any possible mistake here.

Wait, the problem says "the witches have decided to shake hands with each of the sorcerers." So, does that mean that each witch shakes hands with each sorcerer? Or does it mean that each witch shakes hands with each sorcerer once? Well, in handshake problems, usually, each handshake is between two people, and it's counted once. So, if each witch shakes hands with each sorcerer, then it's 6*10=60.

Alternatively, if there was some other restriction, but the problem says that witches don't shake hands with each other, and sorcerers don't shake hands with each other. So, all the handshakes are between witches and sorcerers.

Therefore, the answer is 60.

I think that's solid.

**Final Answer**
The number of handshakes that took place at the mixer is \boxed{60}.

To determine the number of handshakes that occurred at the arcane mixer, we begin by analyzing the structure of the group and the rules governing the handshakes.

---

### **Step 1: Understanding the Participants**

- There are **6 witches** and **10 sorcerers**, making a total of **16 individuals**.
- **Witches** only shake hands with **sorcerers**.
- **Witches do not shake hands with other witches**.
- **Sorcerers do not shake hands with other sorcerers**.

This means that **no handshakes occur within the same group** (witches with witches, or sorcerers with sorcerers), and **all handshakes occur between witches and sorcerers**.

---

### **Step 2: Counting the Handshakes**

Each of the **6 witches** shakes hands with **each of the 10 sorcerers**. Since every witch-sorcerer pair shakes hands exactly once, the total number of handshakes is:

$$
6 \text{ witches} \times 10 \text{ sorcerers} = 60 \text{ handshakes}
$$

This is a straightforward multiplication, as each handshake is uniquely between one witch and one sorcerer.

---

### **Step 3: Alternative Verification via Total Handshakes**

We can also verify this by considering the total number of possible handshakes among all 16 individuals, and then subtracting the ones that **did not occur**.

- **Total possible handshakes** (if everyone shook hands with everyone else):
  $$
  \binom{16}{2} = \frac{16 \times 15}{2} = 120
  $$

- **Handshakes that did not occur**:
  - Between witches: $\binom{6}{2} = 15$
  - Between sorcerers: $\binom{10}{2} = 45$
  - Total non-handshakes: $15 + 45 = 60$

- **Handshakes that did occur**:
  $$
  120 - 60 = 60
  $$

This confirms the earlier result.

---

### **Conclusion**

All methods lead to the same conclusion: the number of handshakes that took place at the mixer is:

$$
\boxed{60}
$$

Solution: The only handshakes at the arcane mixer were between the $6$ witches and $10$ sorcerers, so there were $6 \cdot 10 = \boxed{60}$ handshakes at the mixer.


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:

From 5: cf = 1*1 = 1, correct.

From 4: b*1 + e*1 = b + e = -4. Wait, but from 2, we have b + e = 4. Contradiction. So that's not possible.

Alternatively, c = -1, f = -1. Then:

From 5: (-1)(-1) = 1, correct.

From 4: b*(-1) + e*(-1) = -b - e = -4 ⇒ b + e = 4. Which matches equation 2: b + e = 4. So that's consistent.

So now, 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 factors would be (4x² + 2x -1)(4x² + 2x -1) = (4x² + 2x -1)^2. Let me check if that's correct.

Expanding (4x² + 2x -1)^2:

= (4x²)^2 + (2x)^2 + (-1)^2 + 2*(4x²*2x + 4x²*(-1) + 2x*(-1))

Wait, better to do step by step:

(4x² + 2x -1)(4x² + 2x -1) = 4x²*(4x² + 2x -1) + 2x*(4x² + 2x -1) -1*(4x² + 2x -1)

= 16x^4 + 8x³ -4x² + 8x³ + 4x² -2x -4x² -2x +1

Combine like terms:

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

= 16x^4 + 16x³ -4x² -4x +1. Perfect! That's exactly our quartic. So 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 can solve 4x² + 2x -1 = 0.

Using quadratic formula:

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:

x = [ -1 + sqrt(5) ] / 4 ≈ (-1 + 2.236)/4 ≈ 1.236/4 ≈ 0.309, which is correct because sin(18°) ≈ 0.3090. So yes, that's correct.

Therefore, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait, let me check:

Wait, [ -1 + sqrt(5) ] / 4 is equal to (sqrt(5) - 1)/4. But I thought earlier that sin(18°) is (sqrt(5) - 1)/4 multiplied by something? Wait, let me check with calculator. Let me compute (sqrt(5) - 1)/4:

sqrt(5) ≈ 2.236, so sqrt(5) - 1 ≈ 1.236, divided by 4 ≈ 0.309, which matches sin(18°). So yes, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait, no, it's exactly (sqrt(5) - 1)/4 multiplied by 2? Wait, no, it's (sqrt(5) - 1)/4? Wait, no, let me check:

Wait, if I compute sin(18°), it's approximately 0.3090, and (sqrt(5) - 1)/4 ≈ (2.236 - 1)/4 ≈ 1.236/4 ≈ 0.309, so yes, exactly. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but wait, hold on. Wait, let me check with another formula. I remember that sin(18°) is (sqrt(5)-1)/4 multiplied by 2? Wait, maybe I confused with cos(36°). Let me recall:

cos(36°) is (1 + sqrt(5))/4 * 2? Wait, cos(36°) is (1 + sqrt(5))/4 * 2? Let me check. cos(36°) ≈ 0.8090. Let me compute (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 ≈ 1.618/2 ≈ 0.809, which is correct. Therefore, cos(36°) = (sqrt(5) + 1)/4 * 2? Wait, actually, cos(36°) = [1 + sqrt(5)]/4 * 2? Wait, no, more accurately, cos(36°) = (1 + sqrt(5))/4 * 2? Wait, no, let me write it properly.

From the quadratic solution, we had sin(18°) = [sqrt(5) - 1]/4 * 2? Wait, no, we had sin(18°) = [sqrt(5) - 1]/4? Wait, no, the solution was x = [ -1 + sqrt(5) ] / 4, which is [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, let me check with another approach.

Alternatively, using the formula for sin(18°). Let me recall that in a regular pentagon, the sine of 18 degrees can be related to the golden ratio. But maybe I can just accept that sin(18°) = (sqrt(5) - 1)/4? Wait, but if I compute [sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236/4 ≈ 0.309, which is correct. So yes, sin(18°) = (sqrt(5) - 1)/4? Wait, but actually, I think the exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, let me check with another source. Wait, I can't look it up, but let me think again.

Alternatively, using the identity that sin(18°) = cos(72°). And cos(72°) is known to be (sqrt(5) - 1)/4 * 2? Wait, cos(72°) ≈ 0.3090, same as sin(18°). Wait, but earlier we saw that cos(36°) ≈ 0.8090, which is (sqrt(5) + 1)/4 * 2? Wait, let me check:

Let me recall that cos(36°) = (1 + sqrt(5))/4 * 2? Let me compute (sqrt(5) + 1)/4 * 2 = (sqrt(5) + 1)/2 ≈ (2.236 + 1)/2 ≈ 1.618, which is the golden ratio φ ≈ 1.618, but cos(36°) ≈ 0.809, which is φ/2. Therefore, cos(36°) = φ/2 = (sqrt(5) + 1)/4 * 2? Wait, no:

Wait, φ = (1 + sqrt(5))/2 ≈ 1.618. Therefore, φ/2 ≈ 0.809, which is cos(36°). Therefore, cos(36°) = φ/2 = [1 + sqrt(5)]/4 * 2? Wait, no, cos(36°) = [1 + sqrt(5)]/4 * 2? Wait, [1 + sqrt(5)]/4 * 2 is [1 + sqrt(5)]/2, which is φ. Therefore, cos(36°) = φ/2. Therefore, cos(36°) = [1 + sqrt(5)]/4 * 2? Wait, no, cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, no, cos(36°) = [sqrt(5) + 1]/4 * 2? Let me compute [sqrt(5) + 1]/4 * 2 = [sqrt(5) + 1]/2, which is φ ≈ 1.618, which is greater than 1, which can't be cosine. Wait, so that must be wrong. Wait, so cos(36°) is φ/2 ≈ 0.809, which is [sqrt(5) + 1]/4 * 2? Wait, [sqrt(5) + 1]/4 * 2 is [sqrt(5) + 1]/2, which is φ. So that's not. Wait, so maybe cos(36°) = [sqrt(5) + 1]/4 * 2? No, that would be [sqrt(5) + 1]/2, which is φ. So that's not. Wait, so maybe my previous conclusion was wrong. Wait, let me check:

If we have cos(36°) = 1 - 2sin²(18°). Wait, using the double-angle formula: cos(2θ) = 1 - 2sin²θ. Let me set θ = 18°, then cos(36°) = 1 - 2sin²(18°). Therefore, sin²(18°) = [1 - cos(36°)] / 2. If I can find cos(36°), then I can find sin²(18°). But since we already have sin(18°) = [sqrt(5) - 1]/4? Wait, let me check with that.

If sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309, then sin²(18°) ≈ 0.0955. Then [1 - cos(36°)] / 2 ≈ 0.0955 ⇒ 1 - cos(36°) ≈ 0.191 ⇒ cos(36°) ≈ 1 - 0.191 ≈ 0.809, which is correct. Therefore, that seems okay. Therefore, sin(18°) = [sqrt(5) - 1]/4? Wait, but [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. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with another formula. For example, using the formula for sin(π/10) which is sin(18°). The exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, according to standard references, sin(18°) is (sqrt(5) - 1)/4 * 2? Wait, let me check:

Wait, according to the formula, sin(18°) = (√5 - 1)/4 * 2? Let me compute (√5 - 1)/4 * 2 = (√5 - 1)/2 ≈ (2.236 - 1)/2 ≈ 1.236/2 ≈ 0.618, which is actually 2*sin(18°), since 0.618 is approximately 2*0.309. Wait, so that suggests that sin(18°) = (√5 - 1)/4 * 2? Wait, no, if (√5 - 1)/4 * 2 is (√5 - 1)/2, which is approximately 0.618, which is actually 2*sin(18°). Therefore, sin(18°) = (√5 - 1)/4 * 2? Wait, no, that would be sin(18°) = (√5 - 1)/4 * 2? Wait, no, if (√5 - 1)/4 * 2 is (√5 - 1)/2, which is 2*sin(18°). Therefore, sin(18°) = (√5 - 1)/4? Wait, but that would be half of that. Wait, confusion here. Let me check with exact calculation.

Earlier, we had from solving the equation that sin(18°) = [sqrt(5) - 1]/4? Wait, no, we had x = [sqrt(5) - 1]/4? Wait, no, we had x = [ -1 + sqrt(5) ] / 4, which is [sqrt(5) - 1]/4. So that is sin(18°). But according to the calculation above, if sin(18°) ≈ 0.309, then [sqrt(5) - 1]/4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but according to some sources, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, maybe I need to check with exact value.

Alternatively, let me use another identity. Let me recall that sin(18°) = cos(72°). And cos(72°) can be expressed as (sqrt(5) - 1)/4 * 2? Wait, let me use the formula for cos(72°). Let me use the same method as before. Let me set θ = 36°, then 2θ = 72°, and use double-angle formula. Wait, maybe not. Alternatively, use the same approach as before with multiple angles.

Alternatively, since we have already derived that sin(18°) = [sqrt(5) - 1]/4? Wait, but let me check with decimal approx. sqrt(5) ≈ 2.236, so sqrt(5) - 1 ≈ 1.236, divided by 4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but according to some references, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, if I check with calculator, sin(18°) ≈ 0.3090, and (sqrt(5) - 1)/4 ≈ 0.3090, so that's correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but I thought earlier that cos(36°) = (sqrt(5) + 1)/4 * 2? Wait, no, cos(36°) is [sqrt(5) + 1]/4 * 2? Wait, let me compute [sqrt(5) + 1]/4 * 2 = [sqrt(5) + 1]/2 ≈ (2.236 + 1)/2 ≈ 1.618, which is greater than 1, which is impossible for cosine. Therefore, that must be wrong. Therefore, my previous conclusion that sin(18°) = [sqrt(5) - 1]/4 is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with another formula. For example, using the formula for sin(π/10):

According to standard trigonometric identities, sin(π/10) = sin(18°) = (√5 - 1)/4 * 2? Wait, let me check with exact value. According to the formula here:

Wait, according to the formula, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait, no, according to the formula from the equation we solved, sin(18°) = [sqrt(5) - 1]/4? Wait, but if I check with the exact value, according to the calculation above, that is correct. Therefore, I think that is the correct value. Therefore, moving forward with sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with another approach. Let me use the formula for sin(3θ) or something else.

Alternatively, since we have sin(18°) = [sqrt(5) - 1]/4, then let me compute sin³(18°) + sin²(18°). Let me denote x = sin(18°) = [sqrt(5) - 1]/4. Then compute x³ + x².

But maybe there is a better way. Alternatively, since we have the equation that 4x² + 2x -1 = 0, which came from the quartic equation. Since we know that 4x² + 2x -1 = 0, so 4x² = -2x + 1. Therefore, we can use this to express higher powers of x in terms of lower ones.

Given that, let me compute x³ + x².

First, note that x² = (-2x + 1)/4 from 4x² = -2x + 1.

Therefore, x³ = x * x² = x * [(-2x + 1)/4] = [ -2x² + x ] / 4. But x² is (-2x + 1)/4, so substitute:

x³ = [ -2*(-2x + 1)/4 + x ] / 4 = [ (4x - 2)/4 + x ] / 4 = [ (x - 0.5) + x ] / 4 = [2x - 0.5]/4 = (2x - 1/2)/4 = (4x - 1)/8.

Wait, let me check that again step by step:

Starting with x³ = [ -2x² + x ] / 4.

But x² = (-2x + 1)/4, so substitute:

x³ = [ -2*( (-2x + 1)/4 ) + x ] / 4

= [ (4x - 2)/4 + x ] / 4

Wait, -2*( (-2x + 1)/4 ) = (4x - 2)/4 ?

Yes, because -2*( -2x + 1 ) / 4 = (4x - 2)/4.

Then adding x: (4x - 2)/4 + x = (4x - 2)/4 + 4x/4 = (8x - 2)/4 = (4x - 1)/2.

Therefore, x³ = [ (4x - 1)/2 ] / 4 = (4x - 1)/8.

Therefore, x³ = (4x - 1)/8.

Therefore, x³ + x² = (4x - 1)/8 + x².

But x² = (-2x + 1)/4, so:

x³ + x² = (4x - 1)/8 + (-2x + 1)/4 = convert to eighths:

= (4x - 1)/8 + 2*(-2x + 1)/8 = [4x - 1 -4x + 2]/8 = (1)/8.

Wait, that's interesting! So x³ + x² = 1/8?

Wait, let me check the calculation again:

x³ + x² = (4x - 1)/8 + (-2x + 1)/4.

Convert (-2x + 1)/4 to eighths: multiply numerator and denominator by 2: (-4x + 2)/8.

Therefore, total is (4x - 1 -4x + 2)/8 = ( (4x -4x) + (-1 + 2) ) /8 = (0 + 1)/8 = 1/8.

Wow! So sin³(18°) + sin²(18°) = 1/8? That seems surprisingly simple. Let me check with approximate values to verify.

Compute sin(18°) ≈ 0.3090.

Compute sin³(18°) + sin²(18°) ≈ (0.3090)^3 + (0.3090)^2 ≈ 0.029 + 0.0955 ≈ 0.1245. Now, 1/8 is 0.125. So that's very close. Therefore, the exact value is 1/8. Therefore, the answer is 1/8. Therefore, sin³18° + sin²18° = 1/8.

But let me check again the algebra steps to make sure I didn't make a mistake.

We had:

From 4x² + 2x -1 = 0 ⇒ 4x² = -2x + 1.

Then x³ = x * x² = x * [(-2x + 1)/4] = (-2x² + x)/4.

Then substitute x² again:

= [ -2*(-2x + 1)/4 + x ] / 4

Wait, no, that was:

Wait, x³ = (-2x² + x)/4. Then substitute x² = (-2x + 1)/4:

So:

x³ = [ -2*( (-2x + 1)/4 ) + x ] / 4

= [ (4x - 2)/4 + x ] / 4

Wait, that's:

First term: -2*( (-2x + 1)/4 ) = (4x - 2)/4.

Then adding x: (4x - 2)/4 + x = (4x - 2)/4 + 4x/4 = (8x - 2)/4 = 2x - 0.5.

Then divide by 4: (2x - 0.5)/4 = (4x - 1)/8. Yes, that's correct.

Then x³ + x² = (4x - 1)/8 + x². And x² = (-2x + 1)/4. So:

Convert x² to eighths: (-2x + 1)/4 = (-4x + 2)/8.

Therefore, adding (4x -1)/8 + (-4x + 2)/8 = (4x -1 -4x + 2)/8 = 1/8. Perfect. Therefore, the exact value is 1/8. Therefore, the answer is 1/8.

Therefore, sin³18° + sin²18° = 1/8.

But let me check once more with another approach. Let me compute sin²(18°) + sin³(18°). Let me factor sin²(18°)(1 + sin(18°)). So:

sin²(18°)(1 + sin(18°)).

We know that sin(18°) = (sqrt(5) - 1)/4 ≈ 0.309. Let me compute 1 + sin(18°) ≈ 1.309. Then sin²(18°) ≈ 0.0955. Multiply them: 0.0955 * 1.309 ≈ 0.125, which is 1/8. So that's consistent.

Alternatively, using exact expressions:

Let me compute sin²(18°) * (1 + sin(18°)).

We have sin(18°) = [sqrt(5) - 1]/4. Let me compute 1 + sin(18°) = 1 + [sqrt(5) - 1]/4 = [4 + sqrt(5) - 1]/4 = [3 + sqrt(5)]/4.

Then sin²(18°) = [ (sqrt(5) - 1)/4 ]² = [ (5 - 2sqrt(5) + 1) ] / 16 = [6 - 2sqrt(5)] / 16 = [3 - sqrt(5)] / 8.

Therefore, multiplying sin²(18°) * (1 + sin(18°)) = [3 - sqrt(5)] / 8 * [3 + sqrt(5)] / 4.

Multiply numerators: (3 - sqrt(5))(3 + sqrt(5)) = 9 - (sqrt(5))² = 9 - 5 = 4.

Denominator: 8 * 4 = 32.

Therefore, total is 4 / 32 = 1/8. Perfect! So that confirms it. Therefore, the exact value is indeed 1/8.

Therefore, the answer is 1/8.

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

To compute the value of
$$
\sin^3 18^\circ + \sin^2 18^\circ,
$$
we begin by recognizing that $ \sin 18^\circ $ is a known exact value. Using trigonometric identities and algebraic manipulation, we can derive the exact value of $ \sin 18^\circ $, and then use that to compute the expression.

---

### Step 1: Find the exact value of $ \sin 18^\circ $

We use the identity for $ \sin(5\theta) $, which is:
$$
\sin(5\theta) = 16\sin^5\theta - 20\sin^3\theta + 5\sin\theta.
$$
Let $ x = \sin 18^\circ $. Then $ 5\theta = 90^\circ $, so $ \sin(5\theta) = \sin(90^\circ) = 1 $. Substituting into the identity:
$$
1 = 16x^5 - 20x^3 + 5x.
$$
Rewriting:
$$
16x^5 - 20x^3 + 5x - 1 = 0.
$$
This is a fifth-degree polynomial. Factoring out $ x - 1 $, we find:
$$
(x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0.
$$
Since $ \sin 18^\circ \neq 1 $, we focus on the quartic:
$$
16x^4 + 16x^3 - 4x^2 - 4x + 1 = 0.
$$
Factoring this, we find:
$$
(4x^2 + 2x - 1)^2 = 0.
$$
So the solution is:
$$
4x^2 + 2x - 1 = 0.
$$
Solving this quadratic:
$$
x = \frac{-2 \pm \sqrt{4 + 16}}{8} = \frac{-2 \pm \sqrt{20}}{8} = \frac{-2 \pm 2\sqrt{5}}{8} = \frac{-1 \pm \sqrt{5}}{4}.
$$
Since $ \sin 18^\circ > 0 $, we take the positive root:
$$
\sin 18^\circ = \frac{\sqrt{5} - 1}{4}.
$$

---

### Step 2: Compute $ \sin^3 18^\circ + \sin^2 18^\circ $

Let $ x = \sin 18^\circ = \frac{\sqrt{5} - 1}{4} $. We want to compute:
$$
x^3 + x^2.
$$

From the quadratic equation $ 4x^2 + 2x - 1 = 0 $, we can express $ x^2 $ in terms of $ x $:
$$
4x^2 = -2x + 1 \Rightarrow x^2 = \frac{-2x + 1}{4}.
$$

Now compute $ x^3 $:
$$
x^3 = x \cdot x^2 = x \cdot \frac{-2x + 1}{4} = \frac{-2x^2 + x}{4}.
$$
Substitute $ x^2 = \frac{-2x + 1}{4} $:
$$
x^3 = \frac{-2\left(\frac{-2x + 1}{4}\right) + x}{4} = \frac{\frac{4x - 2}{4} + x}{4} = \frac{4x - 2 + 4x}{16} = \frac{8x - 2}{16} = \frac{4x - 1}{8}.
$$

Now compute $ x^3 + x^2 $:
$$
x^3 + x^2 = \frac{4x - 1}{8} + \frac{-2x + 1}{4}.
$$
Convert the second term to eighths:
$$
x^3 + x^2 = \frac{4x - 1}{8} + \frac{-4x + 2}{8} = \frac{4x - 1 - 4x + 2}{8} = \frac{1}{8}.
$$

---

### 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:

From 5: cf = 1*1 = 1, correct.

From 4: b*1 + e*1 = b + e = -4. Wait, but from 2, we have b + e = 4. Contradiction. So that's not possible.

Alternatively, c = -1, f = -1. Then:

From 5: (-1)(-1) = 1, correct.

From 4: b*(-1) + e*(-1) = -b - e = -4 ⇒ b + e = 4. Which matches equation 2: b + e = 4. So that's consistent.

So now, 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 factors would be (4x² + 2x -1)(4x² + 2x -1) = (4x² + 2x -1)^2. Let me check if that's correct.

Expanding (4x² + 2x -1)^2:

= (4x²)^2 + (2x)^2 + (-1)^2 + 2*(4x²*2x + 4x²*(-1) + 2x*(-1))

Wait, better to do step by step:

(4x² + 2x -1)(4x² + 2x -1) = 4x²*(4x² + 2x -1) + 2x*(4x² + 2x -1) -1*(4x² + 2x -1)

= 16x^4 + 8x³ -4x² + 8x³ + 4x² -2x -4x² -2x +1

Combine like terms:

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

= 16x^4 + 16x³ -4x² -4x +1. Perfect! That's exactly our quartic. So 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 can solve 4x² + 2x -1 = 0.

Using quadratic formula:

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:

x = [ -1 + sqrt(5) ] / 4 ≈ (-1 + 2.236)/4 ≈ 1.236/4 ≈ 0.309, which is correct because sin(18°) ≈ 0.3090. So yes, that's correct.

Therefore, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait, let me check:

Wait, [ -1 + sqrt(5) ] / 4 is equal to (sqrt(5) - 1)/4. But I thought earlier that sin(18°) is (sqrt(5) - 1)/4 multiplied by something? Wait, let me check with calculator. Let me compute (sqrt(5) - 1)/4:

sqrt(5) ≈ 2.236, so sqrt(5) - 1 ≈ 1.236, divided by 4 ≈ 0.309, which matches sin(18°). So yes, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait, no, it's exactly (sqrt(5) - 1)/4 multiplied by 2? Wait, no, it's (sqrt(5) - 1)/4? Wait, no, let me check:

Wait, if I compute sin(18°), it's approximately 0.3090, and (sqrt(5) - 1)/4 ≈ (2.236 - 1)/4 ≈ 1.236/4 ≈ 0.309, so yes, exactly. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but wait, hold on. Wait, let me check with another formula. I remember that sin(18°) is (sqrt(5)-1)/4 multiplied by 2? Wait, maybe I confused with cos(36°). Let me recall:

cos(36°) is (1 + sqrt(5))/4 * 2? Wait, cos(36°) is (1 + sqrt(5))/4 * 2? Let me check. cos(36°) ≈ 0.8090. Let me compute (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 ≈ 1.618/2 ≈ 0.809, which is correct. Therefore, cos(36°) = (sqrt(5) + 1)/4 * 2? Wait, actually, cos(36°) = [1 + sqrt(5)]/4 * 2? Wait, no, more accurately, cos(36°) = (1 + sqrt(5))/4 * 2? Wait, no, let me write it properly.

From the quadratic solution, we had sin(18°) = [sqrt(5) - 1]/4 * 2? Wait, no, we had sin(18°) = [sqrt(5) - 1]/4? Wait, no, the solution was x = [ -1 + sqrt(5) ] / 4, which is [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, let me check with another approach.

Alternatively, using the formula for sin(18°). Let me recall that in a regular pentagon, the sine of 18 degrees can be related to the golden ratio. But maybe I can just accept that sin(18°) = (sqrt(5) - 1)/4? Wait, but if I compute [sqrt(5) - 1]/4 ≈ (2.236 - 1)/4 ≈ 1.236/4 ≈ 0.309, which is correct. So yes, sin(18°) = (sqrt(5) - 1)/4? Wait, but actually, I think the exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, let me check with another source. Wait, I can't look it up, but let me think again.

Alternatively, using the identity that sin(18°) = cos(72°). And cos(72°) is known to be (sqrt(5) - 1)/4 * 2? Wait, cos(72°) ≈ 0.3090, same as sin(18°). Wait, but earlier we saw that cos(36°) ≈ 0.8090, which is (sqrt(5) + 1)/4 * 2? Wait, let me check:

Let me recall that cos(36°) = (1 + sqrt(5))/4 * 2? Let me compute (sqrt(5) + 1)/4 * 2 = (sqrt(5) + 1)/2 ≈ (2.236 + 1)/2 ≈ 1.618, which is the golden ratio φ ≈ 1.618, but cos(36°) ≈ 0.809, which is φ/2. Therefore, cos(36°) = φ/2 = (sqrt(5) + 1)/4 * 2? Wait, no:

Wait, φ = (1 + sqrt(5))/2 ≈ 1.618. Therefore, φ/2 ≈ 0.809, which is cos(36°). Therefore, cos(36°) = φ/2 = [1 + sqrt(5)]/4 * 2? Wait, no, cos(36°) = [1 + sqrt(5)]/4 * 2? Wait, [1 + sqrt(5)]/4 * 2 is [1 + sqrt(5)]/2, which is φ. Therefore, cos(36°) = φ/2. Therefore, cos(36°) = [1 + sqrt(5)]/4 * 2? Wait, no, cos(36°) = [sqrt(5) + 1]/4 * 2? Wait, no, cos(36°) = [sqrt(5) + 1]/4 * 2? Let me compute [sqrt(5) + 1]/4 * 2 = [sqrt(5) + 1]/2, which is φ ≈ 1.618, which is greater than 1, which can't be cosine. Wait, so that must be wrong. Wait, so cos(36°) is φ/2 ≈ 0.809, which is [sqrt(5) + 1]/4 * 2? Wait, [sqrt(5) + 1]/4 * 2 is [sqrt(5) + 1]/2, which is φ. So that's not. Wait, so maybe cos(36°) = [sqrt(5) + 1]/4 * 2? No, that would be [sqrt(5) + 1]/2, which is φ. So that's not. Wait, so maybe my previous conclusion was wrong. Wait, let me check:

If we have cos(36°) = 1 - 2sin²(18°). Wait, using the double-angle formula: cos(2θ) = 1 - 2sin²θ. Let me set θ = 18°, then cos(36°) = 1 - 2sin²(18°). Therefore, sin²(18°) = [1 - cos(36°)] / 2. If I can find cos(36°), then I can find sin²(18°). But since we already have sin(18°) = [sqrt(5) - 1]/4? Wait, let me check with that.

If sin(18°) = [sqrt(5) - 1]/4 ≈ 0.309, then sin²(18°) ≈ 0.0955. Then [1 - cos(36°)] / 2 ≈ 0.0955 ⇒ 1 - cos(36°) ≈ 0.191 ⇒ cos(36°) ≈ 1 - 0.191 ≈ 0.809, which is correct. Therefore, that seems okay. Therefore, sin(18°) = [sqrt(5) - 1]/4? Wait, but [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. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with another formula. For example, using the formula for sin(π/10) which is sin(18°). The exact value is (sqrt(5) - 1)/4 multiplied by 2? Wait, according to standard references, sin(18°) is (sqrt(5) - 1)/4 * 2? Wait, let me check:

Wait, according to the formula, sin(18°) = (√5 - 1)/4 * 2? Let me compute (√5 - 1)/4 * 2 = (√5 - 1)/2 ≈ (2.236 - 1)/2 ≈ 1.236/2 ≈ 0.618, which is actually 2*sin(18°), since 0.618 is approximately 2*0.309. Wait, so that suggests that sin(18°) = (√5 - 1)/4 * 2? Wait, no, if (√5 - 1)/4 * 2 is (√5 - 1)/2, which is approximately 0.618, which is actually 2*sin(18°). Therefore, sin(18°) = (√5 - 1)/4 * 2? Wait, no, that would be sin(18°) = (√5 - 1)/4 * 2? Wait, no, if (√5 - 1)/4 * 2 is (√5 - 1)/2, which is 2*sin(18°). Therefore, sin(18°) = (√5 - 1)/4? Wait, but that would be half of that. Wait, confusion here. Let me check with exact calculation.

Earlier, we had from solving the equation that sin(18°) = [sqrt(5) - 1]/4? Wait, no, we had x = [sqrt(5) - 1]/4? Wait, no, we had x = [ -1 + sqrt(5) ] / 4, which is [sqrt(5) - 1]/4. So that is sin(18°). But according to the calculation above, if sin(18°) ≈ 0.309, then [sqrt(5) - 1]/4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but according to some sources, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, maybe I need to check with exact value.

Alternatively, let me use another identity. Let me recall that sin(18°) = cos(72°). And cos(72°) can be expressed as (sqrt(5) - 1)/4 * 2? Wait, let me use the formula for cos(72°). Let me use the same method as before. Let me set θ = 36°, then 2θ = 72°, and use double-angle formula. Wait, maybe not. Alternatively, use the same approach as before with multiple angles.

Alternatively, since we have already derived that sin(18°) = [sqrt(5) - 1]/4? Wait, but let me check with decimal approx. sqrt(5) ≈ 2.236, so sqrt(5) - 1 ≈ 1.236, divided by 4 ≈ 0.309, which is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but according to some references, sin(18°) is (sqrt(5) - 1)/4 multiplied by 2? Wait, no, if I check with calculator, sin(18°) ≈ 0.3090, and (sqrt(5) - 1)/4 ≈ 0.3090, so that's correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but I thought earlier that cos(36°) = (sqrt(5) + 1)/4 * 2? Wait, no, cos(36°) is [sqrt(5) + 1]/4 * 2? Wait, let me compute [sqrt(5) + 1]/4 * 2 = [sqrt(5) + 1]/2 ≈ (2.236 + 1)/2 ≈ 1.618, which is greater than 1, which is impossible for cosine. Therefore, that must be wrong. Therefore, my previous conclusion that sin(18°) = [sqrt(5) - 1]/4 is correct. Therefore, sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with another formula. For example, using the formula for sin(π/10):

According to standard trigonometric identities, sin(π/10) = sin(18°) = (√5 - 1)/4 * 2? Wait, let me check with exact value. According to the formula here:

Wait, according to the formula, sin(18°) = (sqrt(5) - 1)/4 * 2? Wait, no, according to the formula from the equation we solved, sin(18°) = [sqrt(5) - 1]/4? Wait, but if I check with the exact value, according to the calculation above, that is correct. Therefore, I think that is the correct value. Therefore, moving forward with sin(18°) = (sqrt(5) - 1)/4? Wait, but let me check with another approach. Let me use the formula for sin(3θ) or something else.

Alternatively, since we have sin(18°) = [sqrt(5) - 1]/4, then let me compute sin³(18°) + sin²(18°). Let me denote x = sin(18°) = [sqrt(5) - 1]/4. Then compute x³ + x².

But maybe there is a better way. Alternatively, since we have the equation that 4x² + 2x -1 = 0, which came from the quartic equation. Since we know that 4x² + 2x -1 = 0, so 4x² = -2x + 1. Therefore, we can use this to express higher powers of x in terms of lower ones.

Given that, let me compute x³ + x².

First, note that x² = (-2x + 1)/4 from 4x² = -2x + 1.

Therefore, x³ = x * x² = x * [(-2x + 1)/4] = [ -2x² + x ] / 4. But x² is (-2x + 1)/4, so substitute:

x³ = [ -2*(-2x + 1)/4 + x ] / 4 = [ (4x - 2)/4 + x ] / 4 = [ (x - 0.5) + x ] / 4 = [2x - 0.5]/4 = (2x - 1/2)/4 = (4x - 1)/8.

Wait, let me check that again step by step:

Starting with x³ = [ -2x² + x ] / 4.

But x² = (-2x + 1)/4, so substitute:

x³ = [ -2*( (-2x + 1)/4 ) + x ] / 4

= [ (4x - 2)/4 + x ] / 4

Wait, -2*( (-2x + 1)/4 ) = (4x - 2)/4 ?

Yes, because -2*( -2x + 1 ) / 4 = (4x - 2)/4.

Then adding x: (4x - 2)/4 + x = (4x - 2)/4 + 4x/4 = (8x - 2)/4 = (4x - 1)/2.

Therefore, x³ = [ (4x - 1)/2 ] / 4 = (4x - 1)/8.

Therefore, x³ = (4x - 1)/8.

Therefore, x³ + x² = (4x - 1)/8 + x².

But x² = (-2x + 1)/4, so:

x³ + x² = (4x - 1)/8 + (-2x + 1)/4 = convert to eighths:

= (4x - 1)/8 + 2*(-2x + 1)/8 = [4x - 1 -4x + 2]/8 = (1)/8.

Wait, that's interesting! So x³ + x² = 1/8?

Wait, let me check the calculation again:

x³ + x² = (4x - 1)/8 + (-2x + 1)/4.

Convert (-2x + 1)/4 to eighths: multiply numerator and denominator by 2: (-4x + 2)/8.

Therefore, total is (4x - 1 -4x + 2)/8 = ( (4x -4x) + (-1 + 2) ) /8 = (0 + 1)/8 = 1/8.

Wow! So sin³(18°) + sin²(18°) = 1/8? That seems surprisingly simple. Let me check with approximate values to verify.

Compute sin(18°) ≈ 0.3090.

Compute sin³(18°) + sin²(18°) ≈ (0.3090)^3 + (0.3090)^2 ≈ 0.029 + 0.0955 ≈ 0.1245. Now, 1/8 is 0.125. So that's very close. Therefore, the exact value is 1/8. Therefore, the answer is 1/8. Therefore, sin³18° + sin²18° = 1/8.

But let me check again the algebra steps to make sure I didn't make a mistake.

We had:

From 4x² + 2x -1 = 0 ⇒ 4x² = -2x + 1.

Then x³ = x * x² = x * [(-2x + 1)/4] = (-2x² + x)/4.

Then substitute x² again:

= [ -2*(-2x + 1)/4 + x ] / 4

Wait, no, that was:

Wait, x³ = (-2x² + x)/4. Then substitute x² = (-2x + 1)/4:

So:

x³ = [ -2*( (-2x + 1)/4 ) + x ] / 4

= [ (4x - 2)/4 + x ] / 4

Wait, that's:

First term: -2*( (-2x + 1)/4 ) = (4x - 2)/4.

Then adding x: (4x - 2)/4 + x = (4x - 2)/4 + 4x/4 = (8x - 2)/4 = 2x - 0.5.

Then divide by 4: (2x - 0.5)/4 = (4x - 1)/8. Yes, that's correct.

Then x³ + x² = (4x - 1)/8 + x². And x² = (-2x + 1)/4. So:

Convert x² to eighths: (-2x + 1)/4 = (-4x + 2)/8.

Therefore, adding (4x -1)/8 + (-4x + 2)/8 = (4x -1 -4x + 2)/8 = 1/8. Perfect. Therefore, the exact value is 1/8. Therefore, the answer is 1/8.

Therefore, sin³18° + sin²18° = 1/8.

But let me check once more with another approach. Let me compute sin²(18°) + sin³(18°). Let me factor sin²(18°)(1 + sin(18°)). So:

sin²(18°)(1 + sin(18°)).

We know that sin(18°) = (sqrt(5) - 1)/4 ≈ 0.309. Let me compute 1 + sin(18°) ≈ 1.309. Then sin²(18°) ≈ 0.0955. Multiply them: 0.0955 * 1.309 ≈ 0.125, which is 1/8. So that's consistent.

Alternatively, using exact expressions:

Let me compute sin²(18°) * (1 + sin(18°)).

We have sin(18°) = [sqrt(5) - 1]/4. Let me compute 1 + sin(18°) = 1 + [sqrt(5) - 1]/4 = [4 + sqrt(5) - 1]/4 = [3 + sqrt(5)]/4.

Then sin²(18°) = [ (sqrt(5) - 1)/4 ]² = [ (5 - 2sqrt(5) + 1) ] / 16 = [6 - 2sqrt(5)] / 16 = [3 - sqrt(5)] / 8.

Therefore, multiplying sin²(18°) * (1 + sin(18°)) = [3 - sqrt(5)] / 8 * [3 + sqrt(5)] / 4.

Multiply numerators: (3 - sqrt(5))(3 + sqrt(5)) = 9 - (sqrt(5))² = 9 - 5 = 4.

Denominator: 8 * 4 = 32.

Therefore, total is 4 / 32 = 1/8. Perfect! So that confirms it. Therefore, the exact value is indeed 1/8.

Therefore, the answer is 1/8.

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

To compute the value of
$$
\sin^3 18^\circ + \sin^2 18^\circ,
$$
we begin by recognizing that $ \sin 18^\circ $ is a known exact value. Using trigonometric identities and algebraic manipulation, we can derive the exact value of $ \sin 18^\circ $, and then use that to compute the expression.

---

### Step 1: Find the exact value of $ \sin 18^\circ $

We use the identity for $ \sin(5\theta) $, which is:
$$
\sin(5\theta) = 16\sin^5\theta - 20\sin^3\theta + 5\sin\theta.
$$
Let $ x = \sin 18^\circ $. Then $ 5\theta = 90^\circ $, so $ \sin(5\theta) = \sin(90^\circ) = 1 $. Substituting into the identity:
$$
1 = 16x^5 - 20x^3 + 5x.
$$
Rewriting:
$$
16x^5 - 20x^3 + 5x - 1 = 0.
$$
This is a fifth-degree polynomial. Factoring out $ x - 1 $, we find:
$$
(x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0.
$$
Since $ \sin 18^\circ \neq 1 $, we focus on the quartic:
$$
16x^4 + 16x^3 - 4x^2 - 4x + 1 = 0.
$$
Factoring this, we find:
$$
(4x^2 + 2x - 1)^2 = 0.
$$
So the solution is:
$$
4x^2 + 2x - 1 = 0.
$$
Solving this quadratic:
$$
x = \frac{-2 \pm \sqrt{4 + 16}}{8} = \frac{-2 \pm \sqrt{20}}{8} = \frac{-2 \pm 2\sqrt{5}}{8} = \frac{-1 \pm \sqrt{5}}{4}.
$$
Since $ \sin 18^\circ > 0 $, we take the positive root:
$$
\sin 18^\circ = \frac{\sqrt{5} - 1}{4}.
$$

---

### Step 2: Compute $ \sin^3 18^\circ + \sin^2 18^\circ $

Let $ x = \sin 18^\circ = \frac{\sqrt{5} - 1}{4} $. We want to compute:
$$
x^3 + x^2.
$$

From the quadratic equation $ 4x^2 + 2x - 1 = 0 $, we can express $ x^2 $ in terms of $ x $:
$$
4x^2 = -2x + 1 \Rightarrow x^2 = \frac{-2x + 1}{4}.
$$

Now compute $ x^3 $:
$$
x^3 = x \cdot x^2 = x \cdot \frac{-2x + 1}{4} = \frac{-2x^2 + x}{4}.
$$
Substitute $ x^2 = \frac{-2x + 1}{4} $:
$$
x^3 = \frac{-2\left(\frac{-2x + 1}{4}\right) + x}{4} = \frac{\frac{4x - 2}{4} + x}{4} = \frac{4x - 2 + 4x}{16} = \frac{8x - 2}{16} = \frac{4x - 1}{8}.
$$

Now compute $ x^3 + x^2 $:
$$
x^3 + x^2 = \frac{4x - 1}{8} + \frac{-2x + 1}{4}.
$$
Convert the second term to eighths:
$$
x^3 + x^2 = \frac{4x - 1}{8} + \frac{-4x + 2}{8} = \frac{4x - 1 - 4x + 2}{8} = \frac{1}{8}.
$$

---

### 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).

Wait, let me make sure I didn't make any mistakes here. Let me verify the steps again.

First, the formula for projection is correct? Yes, the projection of a onto b is (a · b / ||b||²) * b.

Then, calculating ||b||²: 2² + 6² + 3² = 4 + 36 + 9 = 49. Correct.

Given that a · b = 8, so substituting into the formula: 8 / 49 times vector b.

Multiplying each component of b by 8/49:

2*(8/49) = 16/49, 6*(8/49) = 48/49, 3*(8/49) = 24/49.

Yes, that seems right.

Alternatively, maybe I can check if there's another way to think about this. For example, if I didn't remember the formula, but I know that the projection scalar is (a · b)/||b||, but wait, no, that's the magnitude of the projection. Wait, no, the scalar projection is (a · b)/||b||, but the vector projection is that scalar times the unit vector in the direction of b.

So, scalar projection is (a · b)/||b||, which would be 8 / sqrt(49) = 8/7. Then the vector projection is that scalar times the unit vector in the direction of b.

Wait, so let me check that. The unit vector in the direction of b is b / ||b||, which is (2/7, 6/7, 3/7). Then multiplying by scalar projection 8/7 gives (8/7 * 2/7, 8/7 * 6/7, 8/7 * 3/7) which is (16/49, 48/49, 24/49).

So same result. Therefore, that confirms that my answer is correct.

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

Alternatively, maybe I can check if there's a different interpretation of the question? The problem says "Find the projection of a onto b", and they gave me a · b = 8. Since they didn't give me vector a, but only the dot product, I think that the formula I used is the correct approach.

Another way to think: if I didn't know the formula, but I know that projection is related to the dot product. Since the dot product is |a||b|cos(theta), where theta is the angle between them. Then the projection of a onto b is |a|cos(theta) times the unit vector in b's direction. Which is (|a|cos(theta)) * (b / |b|).

Which is equal to ( (a · b)/|b|² ) * b. Because |a|cos(theta) is (a · b)/|b|. Therefore, multiplying by b / |b| gives (a · b)/|b|² * b.

So that's the same formula. Therefore, the calculation is correct.

Therefore, I think my answer is correct.

But just to make sure, let me check if I messed up any arithmetic.

Calculating ||b||²: 2² is 4, 6² is 36, 3² is 9. 4 + 36 is 40, plus 9 is 49. Correct.

Then 8 divided by 49 is 8/49. Multiply by each component of b.

2*(8/49) = 16/49. 6*(8/49) = 48/49. 3*(8/49) = 24/49.

Yes, that's correct.

Alternatively, if I wanted to write the vector as fractions, they are already in simplest form since 16, 48, 24 and 49 have no common factors. 49 is 7², and 16 is 2⁴, 48 is 16*3, 24 is 8*3. So no common factors with 49.

Therefore, the answer is (16/49, 48/49, 24/49).

Alternatively, maybe the question expects the answer in some other form? Like a scalar? Wait, no, the projection of a onto b is a vector. The problem says "the projection of a onto b", so it's a vector.

But let me check if maybe they want the scalar projection? But the question says "projection", which usually refers to the vector projection. However, sometimes people refer to scalar projection as the projection. But given that they gave vector b and asked for the projection, it's more likely the vector.

But let me check. If they wanted the scalar projection, it would be (a · b)/||b||, which is 8 / 7. But since they gave vector b, and the answer is a vector, I think the vector projection is expected.

But just to be thorough, let me check both.

If scalar projection: 8 / 7 ≈ 1.142...

But the problem says "projection of a onto b", and in the context of vectors, unless specified otherwise, projection is the vector. Also, since they gave vector b, and the answer is a vector.

Moreover, if they wanted scalar projection, they might have said "the scalar projection" or "the magnitude of the projection".

Therefore, I think my original answer is correct.

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

Just to recap:

proj_b a = (a · b / ||b||²) * b = (8 / 49) * (2, 6, 3) = (16/49, 48/49, 24/49).

Yes, that's correct.

I think that's all. I don't see any mistakes in my reasoning.

**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

$$
\boxed{\begin{pmatrix} \dfrac{16}{49} \\ \dfrac{48}{49} \\ \dfrac{24}{49} \end{pmatrix}}
$$

[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).

Wait, let me make sure I didn't make any mistakes here. Let me verify the steps again.

First, the formula for projection is correct? Yes, the projection of a onto b is (a · b / ||b||²) * b.

Then, calculating ||b||²: 2² + 6² + 3² = 4 + 36 + 9 = 49. Correct.

Given that a · b = 8, so substituting into the formula: 8 / 49 times vector b.

Multiplying each component of b by 8/49:

2*(8/49) = 16/49, 6*(8/49) = 48/49, 3*(8/49) = 24/49.

Yes, that seems right.

Alternatively, maybe I can check if there's another way to think about this. For example, if I didn't remember the formula, but I know that the projection scalar is (a · b)/||b||, but wait, no, that's the magnitude of the projection. Wait, no, the scalar projection is (a · b)/||b||, but the vector projection is that scalar times the unit vector in the direction of b.

So, scalar projection is (a · b)/||b||, which would be 8 / sqrt(49) = 8/7. Then the vector projection is that scalar times the unit vector in the direction of b.

Wait, so let me check that. The unit vector in the direction of b is b / ||b||, which is (2/7, 6/7, 3/7). Then multiplying by scalar projection 8/7 gives (8/7 * 2/7, 8/7 * 6/7, 8/7 * 3/7) which is (16/49, 48/49, 24/49).

So same result. Therefore, that confirms that my answer is correct.

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

Alternatively, maybe I can check if there's a different interpretation of the question? The problem says "Find the projection of a onto b", and they gave me a · b = 8. Since they didn't give me vector a, but only the dot product, I think that the formula I used is the correct approach.

Another way to think: if I didn't know the formula, but I know that projection is related to the dot product. Since the dot product is |a||b|cos(theta), where theta is the angle between them. Then the projection of a onto b is |a|cos(theta) times the unit vector in b's direction. Which is (|a|cos(theta)) * (b / |b|).

Which is equal to ( (a · b)/|b|² ) * b. Because |a|cos(theta) is (a · b)/|b|. Therefore, multiplying by b / |b| gives (a · b)/|b|² * b.

So that's the same formula. Therefore, the calculation is correct.

Therefore, I think my answer is correct.

But just to make sure, let me check if I messed up any arithmetic.

Calculating ||b||²: 2² is 4, 6² is 36, 3² is 9. 4 + 36 is 40, plus 9 is 49. Correct.

Then 8 divided by 49 is 8/49. Multiply by each component of b.

2*(8/49) = 16/49. 6*(8/49) = 48/49. 3*(8/49) = 24/49.

Yes, that's correct.

Alternatively, if I wanted to write the vector as fractions, they are already in simplest form since 16, 48, 24 and 49 have no common factors. 49 is 7², and 16 is 2⁴, 48 is 16*3, 24 is 8*3. So no common factors with 49.

Therefore, the answer is (16/49, 48/49, 24/49).

Alternatively, maybe the question expects the answer in some other form? Like a scalar? Wait, no, the projection of a onto b is a vector. The problem says "the projection of a onto b", so it's a vector.

But let me check if maybe they want the scalar projection? But the question says "projection", which usually refers to the vector projection. However, sometimes people refer to scalar projection as the projection. But given that they gave vector b and asked for the projection, it's more likely the vector.

But let me check. If they wanted the scalar projection, it would be (a · b)/||b||, which is 8 / 7. But since they gave vector b, and the answer is a vector, I think the vector projection is expected.

But just to be thorough, let me check both.

If scalar projection: 8 / 7 ≈ 1.142...

But the problem says "projection of a onto b", and in the context of vectors, unless specified otherwise, projection is the vector. Also, since they gave vector b, and the answer is a vector.

Moreover, if they wanted scalar projection, they might have said "the scalar projection" or "the magnitude of the projection".

Therefore, I think my original answer is correct.

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

Just to recap:

proj_b a = (a · b / ||b||²) * b = (8 / 49) * (2, 6, 3) = (16/49, 48/49, 24/49).

Yes, that's correct.

I think that's all. I don't see any mistakes in my reasoning.

**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

$$
\boxed{\begin{pmatrix} \dfrac{16}{49} \\ \dfrac{48}{49} \\ \dfrac{24}{49} \end{pmatrix}}
$$

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