Find the value of `cos^(-1) (1/2) +2 sin^(-1)(1/2)`

Answer

To find the value of \( \cos^{-1} \left( \frac{1}{2} \right) + 2 \sin^{-1} \left( \frac{1}{2} \right) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \cos^{-1} \left( \frac{1}{2} \right) \)

The value of \( \cos^{-1} \left( \frac{1}{2} \right) \) corresponds to the angle \( \theta \) such that \( \cos \theta = \frac{1}{2} \).

From trigonometric values, we know:
\[ \cos \frac{\pi}{3} = \frac{1}{2} \]
Thus,
\[ \cos^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{3} \]

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

Next, we find \( \sin^{-1} \left( \frac{1}{2} \right) \). The value of \( \sin^{-1} \left( \frac{1}{2} \right) \) corresponds to the angle \( \phi \) such that \( \sin \phi = \frac{1}{2} \).

From trigonometric values, we know:
\[ \sin \frac{\pi}{6} = \frac{1}{2} \]
Thus,
\[ \sin^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{6} \]

### Step 3: Calculate \( 2 \sin^{-1} \left( \frac{1}{2} \right) \)

Now, we compute \( 2 \sin^{-1} \left( \frac{1}{2} \right) \):
\[ 2 \sin^{-1} \left( \frac{1}{2} \right) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

### Step 4: Combine the results

Now we combine 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:
\[ \frac{2\pi}{3} \]

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