# How do you find the exact value of tan^-1 (-1)?

Aug 8, 2017

$- \frac{\pi}{4} \text{ or } \frac{3 \pi}{4}$

#### Explanation:

Well, if

$\arctan \left(- 1\right) = \theta$

then

$\tan \theta = - 1$

There are two values of $\theta$ that satisfy this, according to the unit circle:

color(blue)(ulbar(|stackrel(" ")(" "arctan(-1) = -pi/4 " or " (3pi)/4" ")|)

Aug 8, 2017

${\tan}^{-} 1 x = \theta , x \in \mathbb{R} \iff \tan \theta = x , \theta \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right) .$
Now, $\tan \left(- \frac{\pi}{4}\right) = - \tan \left(\frac{\pi}{4}\right) = - 1 , - \frac{\pi}{4} \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right) .$

$\therefore {\tan}^{-} 1 \left(- 1\right) = - \frac{\pi}{4.}$