How do you find the values of the six trigonometric functions given tantheta=-15/8 and sintheta<0?

Jan 15, 2017

$\sin \theta = - \frac{15}{17}$, $\cos \theta = \frac{8}{17}$, $\tan \theta = - \frac{15}{8}$
$\cot \theta = - \frac{8}{15}$, $\sec \theta = \frac{17}{8}$, $\csc \theta = - \frac{17}{15}$

Explanation:

As $\tan \theta = - \frac{15}{8}$ and $\sin \theta < 0$ i.e. both are negative,

$\theta$ is in Q4 - fourth quadrant and while $\cos \theta$ and $\sec \theta$ are positive, rest of the four trigonometric functions are negative.

Now as $\tan \theta = - \frac{15}{8}$, $\cot \theta = - \frac{8}{15}$ and

$\sec \theta = \sqrt{1 + {\tan}^{2} \theta} = \sqrt{1 + \frac{225}{64}} = \frac{\sqrt{289}}{64} = \frac{17}{8}$

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

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

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