`pi/3-sin^(-1)(-1/2)=?`

Options

A. 0
B. `(2pi)/3`
C. `pi/2`
D. `pi`

Explanation

To solve the expression \( \frac{\pi}{3} - \sin^{-1}(-\frac{1}{2}) \), we can follow these steps:

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

We know that:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Thus, we can write:
\[ \sin^{-1}(-\frac{1}{2}) = -\sin^{-1}(\frac{1}{2}) \]

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

The value of \( \sin^{-1}(\frac{1}{2}) \) is known from trigonometric values:
\[ \sin^{-1}(\frac{1}{2}) = \frac{\pi}{6} \]

### Step 3: Substitute back into the expression

Now substituting back, we have:
\[ \sin^{-1}(-\frac{1}{2}) = -\frac{\pi}{6} \]

### Step 4: Substitute into the original expression

Now we can substitute this value into the original expression:
\[ \frac{\pi}{3} - \sin^{-1}(-\frac{1}{2}) = \frac{\pi}{3} - \left(-\frac{\pi}{6}\right) \]
This simplifies to:
\[ \frac{\pi}{3} + \frac{\pi}{6} \]

### Step 5: Find a common denominator and add

To add these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. Therefore:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
Now we can add:
\[ \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Final Answer

Thus, the value of the expression \( \frac{\pi}{3} - \sin^{-1}(-\frac{1}{2}) \) is:
\[ \frac{\pi}{2} \]

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