`cos^(-1)((1)/2)+2sin ^(-1)((1)/(2))=...`

Options

A. `(pi)/(4)`
B. `(pi)/(6)`
C. `(pi)/(3)`
D. `(2pi)/(3)`

Answer

To solve the expression \( \cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) \), we will follow these steps:

### Step 1: Calculate \( \cos^{-1}\left(\frac{1}{2}\right) \)
The value of \( \cos^{-1}\left(\frac{1}{2}\right) \) corresponds to the angle whose cosine is \( \frac{1}{2} \). This angle is:
\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Step 2: Calculate \( \sin^{-1}\left(\frac{1}{2}\right) \)
The value of \( \sin^{-1}\left(\frac{1}{2}\right) \) corresponds to the angle whose sine is \( \frac{1}{2} \). This angle is:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

### Step 3: Calculate \( 2\sin^{-1}\left(\frac{1}{2}\right) \)
Now we multiply the result from Step 2 by 2:
\[ 2\sin^{-1}\left(\frac{1}{2}\right) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

### Step 4: Combine the results
Now we can add the results from Step 1 and Step 3:
\[ \cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3} \]

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