The value of `sin^(-1){sin(-600^(@))}` is

Options

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

Explanation

To find the value of \( \sin^{-1}(\sin(-600^\circ)) \), we will follow these steps:

### Step 1: Convert degrees to radians
First, we convert \(-600^\circ\) to radians. The conversion formula is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
So,
\[ -600^\circ = -600 \times \frac{\pi}{180} = -\frac{600\pi}{180} = -\frac{10\pi}{3} \]

### Step 2: Simplify the angle
Next, we simplify \(-\frac{10\pi}{3}\) to find an equivalent angle within the range of \([-\pi, \pi]\). We can add \(2\pi\) (which is equivalent to \( \frac{6\pi}{3} \)) to \(-\frac{10\pi}{3}\):
\[ -\frac{10\pi}{3} + 2\pi = -\frac{10\pi}{3} + \frac{6\pi}{3} = -\frac{4\pi}{3} \]

Now, \(-\frac{4\pi}{3}\) is still outside the range of \([- \frac{\pi}{2}, \frac{\pi}{2}]\). We can add another \(2\pi\):
\[ -\frac{4\pi}{3} + 2\pi = -\frac{4\pi}{3} + \frac{6\pi}{3} = \frac{2\pi}{3} \]

### Step 3: Find the sine of the angle
Now we need to find \( \sin\left(-\frac{10\pi}{3}\right) \):
\[ \sin\left(-\frac{10\pi}{3}\right) = \sin\left(\frac{2\pi}{3}\right) \]
Since sine is an odd function:
\[ \sin\left(-\theta\right) = -\sin\left(\theta\right) \]
Thus,
\[ \sin\left(-\frac{10\pi}{3}\right) = -\sin\left(\frac{10\pi}{3}\right) \]
And since \(\frac{10\pi}{3}\) is equivalent to \(\frac{2\pi}{3}\) in sine:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin\left(-\frac{10\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]

### Step 4: Find the inverse sine
Now we find \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \). The value of \( \sin^{-1}(x) \) is defined in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\). The angle whose sine is \(-\frac{\sqrt{3}}{2}\) in this range is:
\[ -\frac{\pi}{3} \]

### Final Answer
Thus, the value of \( \sin^{-1}(\sin(-600^\circ)) \) is:
\[ \boxed{-\frac{\pi}{3}} \]