# How do you use fundamental identities to find the values of the trigonometric values given angle in quadrant 4 such that csc (theta)= -5/2?

Jun 17, 2016

Find values of 6 trig functions

#### Explanation:

$\csc t = - \frac{5}{2}$ --> $\sin t = - \frac{2}{5}$.
Find ${\cos}^{2} t$ by the identity: ${\cos}^{2} t = 1 - {\sin}^{2} t$
${\cos}^{2} t = 1 - \frac{4}{25} = \frac{21}{25}$ --> $\cos t = \pm \frac{\sqrt{21}}{5}$
Since t in in Quadrant IV, then, cos t is positive -->
$\cos t = \frac{\sqrt{21}}{5}$
$\tan t = \frac{\sin}{\cos} = \left(- \frac{2}{5}\right) \left(- \frac{5}{\sqrt{21}}\right) = \frac{2}{\sqrt{21}} = 2 \frac{\sqrt{21}}{21}$
$\cot t = \frac{\sqrt{21}}{2}$
$\sec t = \frac{5}{\sqrt{21}} = 5 \frac{\sqrt{21}}{21}$