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

Aug 24, 2015

for angles in the range $\left[0 , 2 \pi\right]$
${\cos}^{-} 1 \left(- 1\right) = \pi$ (radians)

#### Explanation:

If ${\cos}^{- 1} = \theta$
$\Rightarrow \textcolor{w h i t e}{\text{XXXXX}} \cos \left(\theta\right) = - 1$

This means that the adjacent side is equal in magnitude to the hypotenuse but negative.

Within the range $\left[0 , 2 \pi\right]$
this is only true at $\theta = \pi \left(= {180}^{\circ}\right)$

For all solutions (unrestricted in range:
$\textcolor{w h i t e}{\text{XXX")cos^-1(-1)=pi+n2picolor(white)("XXX}} \forall n \in \mathbb{Z}$