Question
Find the value of:
\[ \cos^{-1}(\cos 1540^\circ) \]
Solution
First, reduce the angle using periodicity:
\[ 1540^\circ = 4 \times 360^\circ + 100^\circ \]
\[ \cos 1540^\circ = \cos 100^\circ \]
Now, we evaluate:
\[ \cos^{-1}(\cos 100^\circ) \]
The principal value range of \( \cos^{-1}x \) is:
\[ [0^\circ, 180^\circ] \]
Since \( 100^\circ \) lies within this range,
\[ \cos^{-1}(\cos 100^\circ) = 100^\circ \]
Final Answer:
\[ \boxed{100^\circ} \]
Key Concept
Always reduce large angles first and then check if the result lies within the principal value range.