Question

Evaluate: $ \tan^{-1} \left[ 2 \sin \left( 2 \cos^{-1} \frac{\sqrt{3}}{2} \right) \right]$

Hint:

When evaluating inverse trigonometric functions, simplify the expression step by step using known values for sine and cosine.

Answer 1

Evaluate $ \cos^{-1} \frac{\sqrt{3}}{2}$. This is equal to $ \frac{\pi}{6}$. Substitute this value into the expression: \[\tan^{-1} \left[ 2 \sin \left( 2 \times \frac{\pi}{6} \right) \right]= \tan^{-1} \left[ 2 \sin \left( \frac{\pi}{3} \right) \right]= \tan^{-1} \left[ 2 \times \frac{\sqrt{3}}{2} \right]= \tan^{-1} (\sqrt{3})\]Since \( \tan^{-1} (\sqrt{3}) = \frac{\pi}{3} \), the result is:\[\frac{\pi}{6}\]