Let `theta=sin^(-1)(sin(-600^@))`, then value of `theta` is:

Options

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

Answer

To solve the problem \( \theta = \sin^{-1}(\sin(-600^\circ)) \), we will follow these steps:

### Step 1: Convert the angle to a standard position
First, we need to convert \(-600^\circ\) into an equivalent angle within the range of \(0^\circ\) to \(360^\circ\). We can do this by adding \(360^\circ\) repeatedly until we get a positive angle.

\[ -600^\circ + 360^\circ \times 2 = -600^\circ + 720^\circ = 120^\circ \]

### Step 2: Find the sine of the angle
Now we can find the sine of \(120^\circ\):

\[ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

### Step 3: Substitute back into the inverse sine function
Now substitute this back into the inverse sine function:

\[ \theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

### Step 4: Determine the angle corresponding to the sine value
The angle whose sine is \(\frac{\sqrt{3}}{2}\) in the range of \([-90^\circ, 90^\circ]\) is:

\[ \theta = 60^\circ \]

### Step 5: Convert the angle back to radians
Finally, we convert \(60^\circ\) to radians:

\[ \theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \]

Thus, the value of \( \theta \) is:

\[ \theta = \frac{\pi}{3} \]

### Final Answer
The value of \( \theta \) is \( \frac{\pi}{3} \).

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