# How do you evaluate arctan(-1) using a unit circle?

Apr 21, 2015

Using a unit circle centered at $\left(0 , 0\right)$ in the Cartesian plane
$\tan \left(\theta\right)$ is the $y$ coordinate value divided by the $x$ coordinate value of the intersection of the unit circle and a ray extending from the origin at an angle of $\theta$

Asking for $\arctan \left(- 1\right)$ is the same as asking to solve
for $\theta$ in
$\tan \left(\theta\right) = - 1$

This will happen when $x$ and $y$ have equal magnitudes but opposite signs
Within the domain $\theta \epsilon \left[0 , 2 \pi\right]$
this only happens at
$\theta = \frac{3 \pi}{4}$
and
$\theta = \frac{7 \pi}{4}$