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

Options

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

Answer

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

### Step 1: Identify the angle
We start with the angle \( \frac{2\pi}{3} \). We need to find the sine of this angle.

### Step 2: Use the sine identity
We can express \( \frac{2\pi}{3} \) in terms of a related angle in the first quadrant. We know that:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \]
This is because \( \frac{2\pi}{3} \) is in the second quadrant where sine is positive.

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

### Step 4: Substitute back into the inverse sine function
Now we substitute back into the inverse sine function:
\[ \sin^{-1}(\sin(\frac{2\pi}{3})) = \sin^{-1}\left(\sin\left(\frac{\pi}{3}\right)\right) \]

### Step 5: Apply the property of inverse sine
The property of inverse sine states that \( \sin^{-1}(\sin x) = x \) if \( x \) is in the range of \( \sin^{-1} \), which is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). However, \( \frac{2\pi}{3} \) is not in this range.

### Step 6: Find the principal value
Since \( \frac{2\pi}{3} \) is outside the range, we need to find the equivalent angle within the range. The equivalent angle is:
\[ \frac{\pi}{3} \]
Thus, we have:
\[ \sin^{-1}(\sin(\frac{2\pi}{3})) = \frac{\pi}{3} \]

### Final Answer
Therefore, the final answer is:
\[ \sin^{-1}(\sin(\frac{2\pi}{3})) = \frac{\pi}{3} \]

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