`sin^(-1) (sin=(2pi)/(3)) = ?`

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(2pi)/(3)`
D. None of these

Answer

To solve the problem \( \sin^{-1}(\sin(\frac{2\pi}{3})) \), we can follow these steps:

### Step 1: Understand the function
The function \( \sin^{-1}(x) \) is the inverse sine function, which gives us an angle whose sine is \( x \). The range of \( \sin^{-1}(x) \) is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

### Step 2: Identify the angle
We start with the angle \( \frac{2\pi}{3} \). We need to find the sine of this angle:
\[ \sin\left(\frac{2\pi}{3}\right) \]

### Step 3: Use the sine identity
Using the identity \( \sin(\pi - x) = \sin(x) \), we can rewrite \( \frac{2\pi}{3} \):
\[ \frac{2\pi}{3} = \pi - \frac{\pi}{3} \]
Thus,
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \]

### Step 4: Calculate sine value
Now we know:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute back into the inverse sine function
Now we can substitute this back into the inverse sine function:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

### Step 6: Find the angle in the range of inverse sine
The angle whose sine is \( \frac{\sqrt{3}}{2} \) that lies within the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) is:
\[ \frac{\pi}{3} \]

### Final Answer
Thus, we have:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) = \frac{\pi}{3} \]

---