# Find the principal values of the following: $tan^{-1} (-1)$

$\begin{array}{1 1} \frac{-\pi}{4} \\ \frac{\pi}{4} \\ \frac{-\pi}{2} \\ \frac{\pi}{2} \end{array}$

Ans : $( tan^{-1}tan \frac{-\pi}{4}=\frac{-\pi}{4} )$
edited Nov 6, 2017

Toolbox:
• The range of the principal value of $\; tan^{-1}x$ is $\left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$
Let $\;tan^{-1}-1= x \Rightarrow tan (x) = -1$
We know that the range of the principal value of $\; tan^{-1}x$ is $\left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$
Therefore, $tan (x) = -1 = - tan \frac{\pi}{4} = tan \frac{-\pi}{4}$
$\Rightarrow x=\frac{-\pi}{4}$, where $x \;\epsilon\;$ $\left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$
Hence the principal value of $x=\frac{-\pi}{4}$
edited Apr 18, 2016 by meena.p