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

Options

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

Explanation

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

### Step 1: Convert the angle to a standard range
The sine function has a periodicity of \( 360^\circ \). To find an equivalent angle for \( -600^\circ \), we can add \( 360^\circ \) multiple times until we get an angle within the standard range of \( [0^\circ, 360^\circ) \).

\[ -600^\circ + 720^\circ = 120^\circ \]

### Step 2: Calculate the sine of the equivalent 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: Apply the inverse sine function
Now we need to find \( \sin^{-1}(\sin(-600^\circ)) \). Since we found that \( \sin(-600^\circ) = \sin(120^\circ) = \frac{\sqrt{3}}{2} \), we can write:

\[ \sin^{-1}(\sin(-600^\circ)) = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

### Step 4: Determine the value of the inverse sine
The value of \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \) corresponds to the angle \( 60^\circ \) in the range \( [-90^\circ, 90^\circ] \).

Thus, we have:

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = 60^\circ \]

### Final Answer
Therefore, the value of \( \sin^{-1}(\sin(-600^\circ)) \) is:

\[ \boxed{60^\circ} \]

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