What is `sin^(-1)"sin"(3 pi)/(5)` equal to ?

Options

A. `(3pi)/(5)`
B. `(2pi)/(5)`
C. `(pi)/(5)`
D. None of these

Explanation

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

### Step 1: Identify the angle
We start with the angle \( \frac{3\pi}{5} \). We know that \( \sin^{-1}(\sin x) \) returns \( x \) if \( x \) is in the range of \( \sin^{-1} \), which is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

### Step 2: Check the range of the angle
The angle \( \frac{3\pi}{5} \) is greater than \( \frac{\pi}{2} \) (which is \( \frac{5\pi}{10} \)), so it is not in the range of \( \sin^{-1} \).

### Step 3: Use the sine function property
We can express \( \frac{3\pi}{5} \) in terms of \( \pi \):
\[ \frac{3\pi}{5} = \pi - \frac{2\pi}{5} \]
Using the sine property \( \sin(\pi - x) = \sin(x) \), we have:
\[ \sin\left(\frac{3\pi}{5}\right) = \sin\left(\frac{2\pi}{5}\right) \]

### Step 4: Apply the inverse sine function
Now we can write:
\[ \sin^{-1}(\sin(3\pi/5)) = \sin^{-1}(\sin(2\pi/5)) \]
Since \( \frac{2\pi}{5} \) is within the range of \( \sin^{-1} \), we can directly conclude:
\[ \sin^{-1}(\sin(2\pi/5)) = \frac{2\pi}{5} \]

### Final Answer
Thus, the value of \( \sin^{-1}(\sin(3\pi/5)) \) is:
\[ \frac{2\pi}{5} \]

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