Directions (FOR NEXT THREE): Read the following information carefully and answer the questions given belaw
Given, `16sin^5x=p sin 5x +qsin 3x + rsin x`
What is the value of q?

Options

A. 3
B. 5
C. 10
D. `-5`

Answer

To find the value of \( q \) in the equation \( 16 \sin^5 x = p \sin 5x + q \sin 3x + r \sin x \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 16 \sin^5 x = p \sin 5x + q \sin 3x + r \sin x \]

2. **Use the identity for \( \sin^5 x \):**
We can express \( \sin^5 x \) in terms of sine functions of multiple angles. We know:
\[ \sin^5 x = \frac{1}{4} (3 \sin x - \sin 3x) \]
Therefore,
\[ 16 \sin^5 x = 16 \cdot \frac{1}{4} (3 \sin x - \sin 3x) = 4 (3 \sin x - \sin 3x) = 12 \sin x - 4 \sin 3x \]

3. **Rewrite the equation:**
Now we substitute this back into our original equation:
\[ 12 \sin x - 4 \sin 3x = p \sin 5x + q \sin 3x + r \sin x \]

4. **Group like terms:**
Rearranging gives us:
\[ p \sin 5x + (q + 4) \sin 3x + (r - 12) \sin x = 0 \]

5. **Equate coefficients:**
For the equation to hold for all \( x \), the coefficients of \( \sin 5x \), \( \sin 3x \), and \( \sin x \) must all be equal to zero:
- Coefficient of \( \sin 5x \): \( p = 0 \)
- Coefficient of \( \sin 3x \): \( q + 4 = 0 \)
- Coefficient of \( \sin x \): \( r - 12 = 0 \)

6. **Solve for \( q \):**
From \( q + 4 = 0 \):
\[ q = -4 \]

### Final Answer:
The value of \( q \) is \( -4 \).