If 15 `cotA=8`, the value of `sinA` is :

Options

A. `(8)/(17)`
B. `(15)/(17)`
C. `(17)/(8)`
D. `(17)/(15)`

Answer

To solve the problem where \( 15 \cot A = 8 \) and we need to find the value of \( \sin A \), we can follow these steps:

### Step 1: Express \( \cot A \) in terms of known values
Given:
\[ 15 \cot A = 8 \]
We can rearrange this to find \( \cot A \):
\[ \cot A = \frac{8}{15} \]

### Step 2: Use the identity of \( \cot A \)
Recall that:
\[ \cot A = \frac{\cos A}{\sin A} \]
From this, we can express \( \cos A \) in terms of \( \sin A \):
\[ \frac{\cos A}{\sin A} = \frac{8}{15} \]
This implies:
\[ \cos A = \frac{8}{15} \sin A \]

### Step 3: Use the Pythagorean identity
We know that:
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting \( \cos A \) from the previous step:
\[ \sin^2 A + \left(\frac{8}{15} \sin A\right)^2 = 1 \]
This simplifies to:
\[ \sin^2 A + \frac{64}{225} \sin^2 A = 1 \]
Combining the terms gives:
\[ \left(1 + \frac{64}{225}\right) \sin^2 A = 1 \]

### Step 4: Simplify the equation
Convert \( 1 \) to a fraction with a denominator of 225:
\[ \frac{225}{225} + \frac{64}{225} = \frac{289}{225} \]
Thus, we have:
\[ \frac{289}{225} \sin^2 A = 1 \]

### Step 5: Solve for \( \sin^2 A \)
To isolate \( \sin^2 A \):
\[ \sin^2 A = \frac{225}{289} \]

### Step 6: Take the square root to find \( \sin A \)
Taking the square root of both sides:
\[ \sin A = \sqrt{\frac{225}{289}} = \frac{15}{17} \]

### Conclusion
Thus, the value of \( \sin A \) is:
\[ \sin A = \frac{15}{17} \]