Question
Find the value of:
\[ \sin^{-1}(\sin (-600^\circ)) \]
Solution
First, reduce the angle:
\[ -600^\circ = -600^\circ + 720^\circ = 120^\circ \]
\[ \sin(-600^\circ) = \sin(120^\circ) \]
Now evaluate:
\[ \sin^{-1}(\sin 120^\circ) \]
The principal value range of \( \sin^{-1}x \) is:
\[ [-90^\circ, 90^\circ] \]
Since \( 120^\circ \) is outside this range, we use identity:
\[ \sin^{-1}(\sin x) = 180^\circ – x \quad \text{for } 90^\circ < x < 180^\circ \]
Thus,
\[ \sin^{-1}(\sin 120^\circ) = 180^\circ – 120^\circ = 60^\circ \]
Final Answer:
\[ \boxed{60^\circ} \]
Key Concept
Always convert the angle into one full cycle and then apply principal value rules.