# How do you find the values of the six trigonometric functions given costheta=8/17 and tantheta<0?

Jan 7, 2017

$\cos \theta = \frac{8}{17}$

$\sin \theta = - \frac{15}{17}$

$\tan \theta = - \frac{15}{8}$

$\cot \theta = - \frac{8}{15}$

$\sec \theta = \frac{17}{8}$

$\csc \theta = - \frac{17}{15}$

#### Explanation:

Since

$\cos \theta > 0$

and

$\tan \theta < 0$,

the angle is in the fourth quadrant, where

$\sin \theta < 0$,

then
$\sin \theta = - \sqrt{1 - {\cos}^{2} \theta} = - \sqrt{1 - {\left(\frac{8}{17}\right)}^{2}} = - \sqrt{1 - \frac{64}{289}} = - \sqrt{\frac{225}{289}} = - \frac{15}{17}$

$\tan \theta = \sin \frac{\theta}{\cos} \theta = \frac{- \frac{15}{\cancel{17}}}{\frac{8}{\cancel{17}}} = - \frac{15}{8}$

$\cot \theta = \frac{1}{\tan} \theta = - \frac{8}{15}$

$\sec \theta = \frac{1}{\cos} \theta = \frac{17}{8}$

$\csc \theta = \frac{1}{\sin} \theta = - \frac{17}{15}$