The value of `sin^(-1)[cos{sin^(-1)(-sqrt(3)/(2))}]` is

Options

A. `(pi)/(3)`
B. `(pi)/(6)`
C. `-(pi)/(3)`
D. `-(pi)/(6)`

Explanation

To solve the problem \( \sin^{-1}[\cos(\sin^{-1}(-\sqrt{3}/2))] \), we will break it down step by step.

### Step 1: Find \( \sin^{-1}(-\sqrt{3}/2) \)

The value of \( \sin^{-1}(-\sqrt{3}/2) \) corresponds to an angle where the sine is \(-\sqrt{3}/2\). The angle that has a sine of \(\sqrt{3}/2\) is \(\pi/3\). Since we are looking for the negative value, we have:

\[ \sin^{-1}(-\sqrt{3}/2) = -\frac{\pi}{3} \]

### Step 2: Substitute into the cosine function

Now we substitute this value into the cosine function:

\[ \cos(\sin^{-1}(-\sqrt{3}/2)) = \cos\left(-\frac{\pi}{3}\right) \]

Using the property of cosine, where \(\cos(-\theta) = \cos(\theta)\), we get:

\[ \cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \]

### Step 3: Calculate \( \cos(\pi/3) \)

The value of \( \cos(\pi/3) \) is known to be:

\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 4: Find \( \sin^{-1}(\frac{1}{2}) \)

Now we need to find:

\[ \sin^{-1}\left(\frac{1}{2}\right) \]

The angle whose sine is \(\frac{1}{2}\) is:

\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

### Final Answer

Thus, the value of \( \sin^{-1}[\cos(\sin^{-1}(-\sqrt{3}/2))] \) is:

\[ \frac{\pi}{6} \]