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Question

Mathematics Question on Properties of Inverse Trigonometric Functions

tan⁡−1(tan⁡3π4)\tan^{-1}\left(\tan\frac{3\pi}{4}\right) is equal to

A

π4\frac{\pi}{4}

B

−π4-\frac{\pi}{4}

C

3π4\frac{3\pi}{4}

D

none of these.

Answer

−π4-\frac{\pi}{4}

Explanation

Solution

tan−1(tan3π4)=tan−1(tan(π−π4))tan^{-1} \left(tan \frac{3\pi}{4}\right) = tan^{-1} \left(tan \left(\pi-\frac{\pi}{4}\right)\right) =tan−1(−tanπ4)= tan^{-1}\left(-tan \frac{\pi}{4}\right) =tan−1(tan(−π4)) = tan^{-1} \left(tan\left(-\frac{\pi}{4}\right)\right) =−π4= -\frac{\pi}{4} [∵tan−1(tan θ)=θ if−π2≤θ≤π2]\left[\because tan^{-1}\left(tan\,\theta \right)= \theta\, {\text{if}} -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\right]