# How do you find the values of the six trigonometric functions given csctheta=4 and cottheta<0?

Feb 25, 2018

$\sin \theta = \frac{1}{4}$, $\cos \theta = - \frac{\sqrt{15}}{4}$, $\tan \theta = - \frac{1}{\sqrt{15}}$
$\cot \theta = - \sqrt{15}$, $\sec \theta = - \frac{4}{\sqrt{15}}$ and $\csc \theta = 4$.

#### Explanation:

Let us consider the identity ${\csc}^{2} \theta = 1 + {\cot}^{2} \theta$

as $\csc \theta = 4$, we have $16 = 1 + {\cot}^{2} \theta$ and

${\cot}^{2} \theta = 15$ and as $\cot \theta < 0$, $\cot \theta = - \sqrt{15}$

Therefore $\tan \theta = - \frac{1}{\sqrt{15}}$ and $\sin \theta = \frac{1}{4}$

As #tantheta=sintheta/costheta, this means

$\cos \theta = \sin \frac{\theta}{\tan} \theta = \frac{\frac{1}{4}}{- \frac{1}{\sqrt{15}}} = - \frac{\sqrt{15}}{4}$

and $\sec \theta = - \frac{4}{\sqrt{15}}$