If `cos^(-1) x = (pi)/(3)`, the find the value of `sin ^(-1)x`.

Answer

To solve the problem, we start with the equation given:

\[ \cos^{-1} x = \frac{\pi}{3} \]

### Step 1: Find the value of \( x \)

From the equation \( \cos^{-1} x = \frac{\pi}{3} \), we can find \( x \) by taking the cosine of both sides:

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

### Step 2: Calculate \( \cos\left(\frac{\pi}{3}\right) \)

We know that:

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

Thus, we have:

\[ x = \frac{1}{2} \]

### Step 3: Find \( \sin^{-1} x \)

Now, we need to find \( \sin^{-1} x \):

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

### Step 4: Calculate \( \sin^{-1}\left(\frac{1}{2}\right) \)

We know that:

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

Thus, we have:

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

### Final Answer

Therefore, the value of \( \sin^{-1} x \) is:

\[ \sin^{-1} x = \frac{\pi}{6} \]

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