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

Options

A. `-sqrt(3)/(2)`
B. `-(1)/(2)`
C. `(1)/(2)`
D. `sqrt(3)/(2)`

Answer

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

### Step 1: Rewrite the Inverse Sine
We start with the expression:
\[ \sin\left(\sin^{-1}\left(-\frac{1}{2}\right) + \frac{\pi}{3}\right) \]
Using the property of inverse sine, we know that:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Thus, we can rewrite:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\sin^{-1}\left(\frac{1}{2}\right) \]

### Step 2: Find the Value of \( \sin^{-1}\left(\frac{1}{2}\right) \)
We know that:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
Therefore:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]

### Step 3: Substitute Back into the Expression
Now we can substitute this back into our original expression:
\[ \sin\left(-\frac{\pi}{6} + \frac{\pi}{3}\right) \]

### Step 4: Simplify the Angle
To simplify the angle, we need a common denominator:
\[ -\frac{\pi}{6} + \frac{\pi}{3} = -\frac{\pi}{6} + \frac{2\pi}{6} = \frac{\pi}{6} \]

### Step 5: Calculate the Sine
Now we find:
\[ \sin\left(\frac{\pi}{6}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

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

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