# How do you find the exact value of  cot(arcsin ((-7/13))?

Jun 17, 2018

$\pm 2 \frac{\sqrt{30}}{7}$

#### Explanation:

it depends on how you define range of $\arcsin \left(\theta\right)$
1. range of $\arcsin \left(\theta\right) = \left\{y | 0 < y \le \frac{\pi}{2} , \pi < y \le \frac{3}{2} \cdot \pi\right\}$
The angle of $\arcsin \left(- \frac{7}{13}\right)$ is at third quadrant.
$\cot \left(\arcsin \left(- \frac{7}{13}\right)\right) = - \frac{\sqrt{{13}^{2} - {\left(- 7\right)}^{2}}}{- 7} = 2 \frac{\sqrt{30}}{7}$
2. range of $\arcsin \left(\theta\right) = \left\{y | - \frac{\pi}{2} \le y < 0 , 0 < y \le \frac{\pi}{2}\right\}$
The angle of $\arcsin \left(- \frac{7}{13}\right)$ is at fourth quadrant.
$\cot \left(\arcsin \left(- \frac{7}{13}\right)\right) = \frac{\sqrt{{13}^{2} - {\left(- 7\right)}^{2}}}{- 7} = - 2 \frac{\sqrt{30}}{7}$