The value of `sin(cos^(-1)3/5)` is

Options

A. `2/5`
B. `4/5`
C. `-2/5`
D. none of these

Explanation

To find the value of \( \sin(\cos^{-1} \frac{3}{5}) \), we can follow these steps:

### Step 1: Understand the relationship between cosine and sine
We know that if \( \theta = \cos^{-1}(x) \), then \( \cos(\theta) = x \). In our case, we have:
\[ \theta = \cos^{-1}\left(\frac{3}{5}\right) \]
This means:
\[ \cos(\theta) = \frac{3}{5} \]

### Step 2: Use the Pythagorean identity
Using the Pythagorean identity, we can find \( \sin(\theta) \):
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting \( \cos(\theta) \):
\[ \sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1 \]
This simplifies to:
\[ \sin^2(\theta) + \frac{9}{25} = 1 \]

### Step 3: Solve for \( \sin^2(\theta) \)
Rearranging the equation gives:
\[ \sin^2(\theta) = 1 - \frac{9}{25} \]
\[ \sin^2(\theta) = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

### Step 4: Find \( \sin(\theta) \)
Taking the square root of both sides, we find:
\[ \sin(\theta) = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
Since \( \theta = \cos^{-1}(\frac{3}{5}) \) is in the range \( [0, \pi] \), \( \sin(\theta) \) is non-negative.

### Conclusion
Thus, the value of \( \sin(\cos^{-1} \frac{3}{5}) \) is:
\[ \frac{4}{5} \]