Find the value of `sin^(-1)("sin"(3pi)/5)`

Options

A. `(2pi)/5`
B. `pi/4`
C. `(3pi)/5`
D. `pi/2`

Answer

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

### Step 1: Identify the angle
We start with the angle \( \frac{3\pi}{5} \).

### Step 2: Determine the sine value
Calculate \( \sin\left(\frac{3\pi}{5}\right) \). The sine function is positive in the second quadrant, where \( \frac{3\pi}{5} \) lies.

### Step 3: Use the sine inverse function
The sine inverse function \( \sin^{-1}(x) \) gives us an angle in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). Since \( \frac{3\pi}{5} \) is not in this range, we need to find an equivalent angle that is.

### Step 4: Find the equivalent angle
The sine function is periodic, and we can use the property that:
\[ \sin(\theta) = \sin(\pi - \theta) \]
Thus, we can find an angle in the range of \( \sin^{-1} \) that has the same sine value:
\[ \sin\left(\frac{3\pi}{5}\right) = \sin\left(\pi - \frac{3\pi}{5}\right) = \sin\left(\frac{2\pi}{5}\right) \]

### Step 5: Calculate the sine inverse
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 \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we have:
\[ \sin^{-1}(\sin(2\pi/5)) = \frac{2\pi}{5} \]

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