# How do you calculate cos^-1 (-0.8)?

Dec 16, 2015

$\theta = {143.13}^{\circ}$or ${216.87}^{\circ}$

#### Explanation:

Since -0.8 is not a value you would find in a trigonometric ratio, which are: $\sin \theta$, $\cos \theta$, $\tan \theta$, $\csc \theta$, $\sec \theta$, $\cot \theta$, you have to use a calculator to find the answer:

$\cos \theta = - 0.8$
$\textcolor{w h i t e}{\times} \theta = {\cos}^{-} 1 \left(- 0.8\right)$
$\textcolor{w h i t e}{\times x} \approx {143.13}^{\circ}$

However, according to the CAST rule, cosine is also negative in the third quadrant, which means there is a second possible answer.

First, find the related acute angle, and add the angle to ${180}^{\circ}$ to find the second possible answer.

Finding the related acute angle
${180}^{\circ} - {143.13}^{\circ}$
$= {36.87}^{\circ}$

Finding the second possible angle
$= {180}^{\circ} + {36.87}^{\circ}$
$= {216.87}^{\circ}$

$\therefore$, the two possible angles are ${143.13}^{\circ}$ and ${216.87}^{\circ}$.