Question
Find the value of:
\[ \sin^{-1}(\sin 1550^\circ) \]
Solution
First, reduce the angle:
\[ 1550^\circ = 4 \times 360^\circ + 110^\circ \]
\[ \sin 1550^\circ = \sin 110^\circ \]
Now evaluate:
\[ \sin^{-1}(\sin 110^\circ) \]
The principal value range of \( \sin^{-1}x \) is:
\[ [-90^\circ, 90^\circ] \]
Since \( 110^\circ \) is outside this range, we use identity:
\[ \sin^{-1}(\sin x) = 180^\circ – x \quad \text{for } 90^\circ < x < 180^\circ \]
Thus,
\[ \sin^{-1}(\sin 110^\circ) = 180^\circ – 110^\circ = 70^\circ \]
Final Answer:
\[ \boxed{70^\circ} \]
Key Concept
Reduce the angle and apply principal value rules carefully.