# How do you use tantheta=4 to find csctheta?

Jan 14, 2017

Use the identities $\cot \left(\theta\right) = \frac{1}{\tan} \left(\theta\right)$ and $\csc \left(\theta\right) = \pm \sqrt{1 + {\cot}^{2} \left(\theta\right)}$

#### Explanation:

$\cot \left(\theta\right) = \frac{1}{\tan} \left(\theta\right) = \frac{1}{4}$

$\csc \left(\theta\right) = \pm \sqrt{1 + {\cot}^{2} \left(\theta\right)}$

$\csc \left(\theta\right) = \pm \sqrt{1 + {\left(\frac{1}{4}\right)}^{2}}$

$\csc \left(\theta\right) = \pm \sqrt{1 + \frac{1}{16}}$

$\csc \left(\theta\right) = \pm \frac{\sqrt{17}}{4}$

It is impossible to tell whether to use the plus or minus case, without knowing whether the angle is in the first or the third quadrant.