Value of `cos^(-1) (-(1)/(2)) + sin ^(-1)""(1)/(2)` is: a) `2 pi` b) `(pi)/(2)` c) `pi` d) none of these

Options

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

Answer

To solve the expression \( \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(\frac{1}{2}) \), we will break it down step by step.

### Step 1: Evaluate \( \cos^{-1}(-\frac{1}{2}) \)
The cosine function is negative in the second quadrant. The angle whose cosine is \(-\frac{1}{2}\) is \( \frac{2\pi}{3} \) (or \( 120^\circ \)). Therefore:
\[ \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \]

### Step 2: Evaluate \( \sin^{-1}(\frac{1}{2}) \)
The sine function is positive in the first quadrant. The angle whose sine is \(\frac{1}{2}\) is \( \frac{\pi}{6} \) (or \( 30^\circ \)). Therefore:
\[ \sin^{-1}(\frac{1}{2}) = \frac{\pi}{6} \]

### Step 3: Add the two results
Now we add the two angles we found:
\[ \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(\frac{1}{2}) = \frac{2\pi}{3} + \frac{\pi}{6} \]

### Step 4: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. We convert \( \frac{2\pi}{3} \) to have a denominator of 6:
\[ \frac{2\pi}{3} = \frac{4\pi}{6} \]

Now we can add:
\[ \frac{4\pi}{6} + \frac{\pi}{6} = \frac{4\pi + \pi}{6} = \frac{5\pi}{6} \]

### Final Answer
Thus, the value of \( \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(\frac{1}{2}) \) is:
\[ \frac{5\pi}{6} \]

Since \( \frac{5\pi}{6} \) is not listed among the options a) \( 2\pi \), b) \( \frac{\pi}{2} \), c) \( \pi \), d) none of these, the correct choice is:
**d) none of these.**