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Question

Mathematics Question on Inverse Trigonometric Functions

A value of tan−1[sin(cos−1(23))]tan^{-1}\left[sin\left(cos^{-1}\left(\sqrt{\frac{2}{3}}\right)\right)\right] is

A

π4\frac{\pi}{4}

B

π2\frac{\pi}{2}

C

π3\frac{\pi}{3}

D

π6\frac{\pi}{6}

Answer

π6\frac{\pi}{6}

Explanation

Solution

Consider tan−1[sin(cos−123)]tan^{-1}\left[sin\left(cos^{-1}\sqrt{\frac{2}{3}}\right)\right] Let cos−123=θ⇒cos θ=23cos^{-1}\sqrt{\frac{2}{3}}=\theta \Rightarrow cos\,\theta=\sqrt{\frac{2}{3}} ⇒sin θ=1−cos2 θ=1−23=13\Rightarrow sin\,\theta=\sqrt{1-cos^{2}\,\theta}=\sqrt{1-\frac{2}{3}}=\sqrt{\frac{1}{3}} ∴tan−1[sin(cos−123)]=tan−1[sin θ]\therefore tan^{-1}\left[sin\left(cos^{-1}\sqrt{\frac{2}{3}}\right)\right]=tan^{-1}\left[sin\,\theta\right] =tan−1[13]=tan−1(13)=tan^{-1}\left[\sqrt{\frac{1}{3}}\right]=tan^{-1}\left(\frac{1}{\sqrt{3}}\right) =π6= \frac{\pi}{6}