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

Oct 28, 2016

#### Explanation:

The inverse tangent function will return a negative angle for this where $\cos \left(\theta\right) = - \sin \left(\theta\right)$. The angle that it returned is:

$\theta = - \frac{\pi}{4}$

Doing this causes a loss of information, because the condition $\cos \left(\theta\right) = - \sin \left(\theta\right)$ occurs in the second AND the fourth quadrant. If you have additional information, regarding which quadrant, you should to one of the two following computations:

$\theta = {\tan}^{-} 1 \left(- 1\right) + \pi$

$\theta = - \frac{\pi}{4} + \pi$

$\theta = \frac{3 \pi}{4}$

$\theta = {\tan}^{-} 1 \left(- 1\right) + 2 \pi$
$\theta = - \frac{\pi}{4} + 2 \pi$
$\theta = \frac{7 \pi}{4}$