# How do you evaluate cot^-1(-5)?

Dec 12, 2015

${348.69}^{\circ}$ or ${168.69}^{\circ}$

#### Explanation:

$\theta = {\cot}^{-} 1 \left(- 5\right)$
$\cot \theta = - 5$
$\frac{1}{\tan} \theta = - 5$
$\tan \theta = - \frac{1}{5}$
$\theta = {\tan}^{-} 1 \left(- \frac{1}{5}\right)$
$\theta \approx - {11.31}^{\circ} \left({348.69}^{\circ}\right)$

However, according to the CAST rule, $\cot \theta$ and $\tan \theta$ are also negative in second quadrant.

To find the second possible angle, find the related acute angle, and subtract the value from ${180}^{\circ}$.

Finding the related acute angle:
${360}^{\circ} - {348.69}^{\circ}$
$= {11.31}^{\circ}$

Finding the second possible angle:
${180}^{\circ} - {11.31}^{\circ}$
$= {168.69}^{\circ}$

$\therefore$, the possible angles are approximately ${348.69}^{\circ}$ and ${168.69}^{\circ}$.