Question
If
\[ \theta = \sin^{-1}(\sin(-600^\circ)) \]
Find one possible value of \( \theta \).
Solution
First reduce the angle:
\[ -600^\circ = -600^\circ + 720^\circ = 120^\circ \]
\[ \sin(-600^\circ) = \sin(120^\circ) \]
\[ = \sin(180^\circ – 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Now,
\[ \theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
Principal value range of \( \sin^{-1}x \):
\[ [-90^\circ, 90^\circ] \]
So,
\[ \theta = 60^\circ \]
Final Answer:
\[ \boxed{60^\circ} \]
Key Concept
Reduce the angle first and then apply principal value range.