`sin[3sin^(-1)((1)/(5))]` is equal to

Options

A. `(71)/(125)`
B. `(74)/(125)`
C. `(3)/(5)`
D. `(1)/(2)`

Explanation

To solve the expression \( \sin\left(3 \sin^{-1}\left(\frac{1}{5}\right)\right) \), we can use the formula for \( \sin(3\theta) \) in terms of \( \sin(\theta) \).

### Step-by-Step Solution:

1. **Identify the value of \( x \)**:
Let \( x = \sin^{-1}\left(\frac{1}{5}\right) \). Therefore, we have:
\[ \sin(x) = \frac{1}{5} \]

2. **Use the formula for \( \sin(3\theta) \)**:
The formula for \( \sin(3\theta) \) is:
\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \]
Here, we will substitute \( \theta = x \).

3. **Substitute \( \sin(x) \)**:
Now substituting \( \sin(x) = \frac{1}{5} \):
\[ \sin(3x) = 3\left(\frac{1}{5}\right) - 4\left(\frac{1}{5}\right)^3 \]

4. **Calculate \( \left(\frac{1}{5}\right)^3 \)**:
\[ \left(\frac{1}{5}\right)^3 = \frac{1}{125} \]

5. **Substitute this value back**:
\[ \sin(3x) = 3\left(\frac{1}{5}\right) - 4\left(\frac{1}{125}\right) \]
\[ = \frac{3}{5} - \frac{4}{125} \]

6. **Find a common denominator**:
The common denominator between 5 and 125 is 125. Convert \( \frac{3}{5} \) to have a denominator of 125:
\[ \frac{3}{5} = \frac{3 \times 25}{5 \times 25} = \frac{75}{125} \]

7. **Combine the fractions**:
Now we can combine the two fractions:
\[ \sin(3x) = \frac{75}{125} - \frac{4}{125} = \frac{75 - 4}{125} = \frac{71}{125} \]

8. **Final Result**:
Thus, the value of \( \sin\left(3 \sin^{-1}\left(\frac{1}{5}\right)\right) \) is:
\[ \frac{71}{125} \]