Question
Find the value of:
\[ \cos^{-1}(\cos 350^\circ) – \sin^{-1}(\sin 350^\circ) \]
Solution
We use principal value ranges:
- \( \cos^{-1}x \in [0^\circ, 180^\circ] \)
- \( \sin^{-1}x \in [-90^\circ, 90^\circ] \)
First,
\[ \cos^{-1}(\cos 350^\circ) \]
Since \( 350^\circ = 360^\circ – 10^\circ \),
\[ \cos 350^\circ = \cos 10^\circ \]
Thus,
\[ \cos^{-1}(\cos 350^\circ) = \cos^{-1}(\cos 10^\circ) \]
Since \( 10^\circ \in [0^\circ, 180^\circ] \),
\[ = 10^\circ \]
Next,
\[ \sin^{-1}(\sin 350^\circ) \]
Since \( 350^\circ = 360^\circ – 10^\circ \),
\[ \sin 350^\circ = -\sin 10^\circ \]
So,
\[ \sin^{-1}(\sin 350^\circ) = \sin^{-1}(-\sin 10^\circ) \]
Since \( -10^\circ \in [-90^\circ, 90^\circ] \),
\[ = -10^\circ \]
Therefore,
\[ \cos^{-1}(\cos 350^\circ) – \sin^{-1}(\sin 350^\circ) = 10^\circ – (-10^\circ) \]
\[ = 20^\circ \]
Final Answer:
\[ \boxed{20^\circ} \]
Key Concept
Reduce angles and carefully apply principal value ranges to avoid mistakes.