Write down the value of `2 sin^(-1) . 3/5`

Answer

To find the value of \( 2 \sin^{-1} \left( \frac{3}{5} \right) \), we can follow these steps:

### Step 1: Let \( \theta = \sin^{-1} \left( \frac{3}{5} \right) \)
This means that \( \sin \theta = \frac{3}{5} \).

### Step 2: Find \( \cos \theta \)
Using the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta = \frac{3}{5} \):
\[ \left( \frac{3}{5} \right)^2 + \cos^2 \theta = 1 \]
\[ \frac{9}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]
Taking the square root:
\[ \cos \theta = \frac{4}{5} \]

### Step 3: Find \( \sin(2\theta) \)
Using the double angle formula:
\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]
Substituting the values we found:
\[ \sin(2\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} \]
Calculating:
\[ \sin(2\theta) = \frac{24}{25} \]

### Step 4: Conclusion
Thus, the value of \( 2 \sin^{-1} \left( \frac{3}{5} \right) \) is:
\[ \sin(2\theta) = \frac{24}{25} \]

### Final Answer:
\[ 2 \sin^{-1} \left( \frac{3}{5} \right) = \sin^{-1} \left( \frac{24}{25} \right) \]

---