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Question

Mathematics Question on Inverse Trigonometric Functions

Find the principal value tan−1 (−1)tan^{-1} \, (-1)

A

2π3\frac{2\pi}{3}

B

π10\frac{\pi}{10}

C

3π4\frac{3\pi}{4}

D

−π4-\frac{\pi}{4}

Answer

−π4-\frac{\pi}{4}

Explanation

Solution

Let tan−1(−1)=θtan^{-1} \left(-1\right) = \theta ⇒tan θ=−1\Rightarrow tan\,\theta = -1 =−tanπ4= -tan \frac{\pi}{4} =tan(−π4)= tan \left(-\frac{\pi }{4}\right) ⇒θ=−π4∈(−π2,π2)\Rightarrow \theta = -\frac{\pi }{4} \in \left(\frac{-\pi }{2}, \frac{\pi }{2}\right) ∴\therefore Principal value of tan−1(−1)tan^{-1} \left(-1\right) is −π4\frac{-\pi }{4}