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

Options

A. A) 5
B. B) 8
C. C) 10
D. D) -10

Answer

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

### Step 1: Rewrite \( \sin^5 x \)
We start with the left-hand side of the equation:
\[ 16 \sin^5 x \]
Using the identity for \( \sin^5 x \), we can express it in terms of \( \sin 3x \) and \( \sin x \). We know that:
\[ \sin^5 x = \frac{1}{16} \left( 5 \sin x - 5 \sin 3x + \sin 5x \right) \]
Thus, we can rewrite:
\[ 16 \sin^5 x = 5 \sin x - 5 \sin 3x + \sin 5x \]

### Step 2: Expand the Left Side
Now, substituting this back into our equation gives us:
\[ 16 \sin^5 x = 5 \sin x - 5 \sin 3x + \sin 5x \]

### Step 3: Compare with the Right Side
Now we compare this with the right-hand side of the original equation:
\[ p \sin 5x + q \sin 3x + r \sin x \]
From our expansion, we have:
\[ \sin 5x + (-5) \sin 3x + 5 \sin x \]
This implies:
- Coefficient of \( \sin 5x \): \( p = 1 \)
- Coefficient of \( \sin 3x \): \( q = -5 \)
- Coefficient of \( \sin x \): \( r = 5 \)

### Step 4: Conclusion
From the comparison, we find that:
\[ r = 5 \]

### Final Answer
Thus, the value of \( r \) is \( 5 \).

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