# If $\sec x=-2$ and $\pi < x < \large\frac{3\pi}{2}$,find the value of all other five trigonometric functions.

Toolbox:
• $\sin ^2\theta+\cos ^2\theta=1$
• $\sin \theta=\sqrt{1-\cos ^2\theta}$
• $\tan ^2\theta+1=\sec^2\theta$
• $\tan \theta=\sqrt{\sec^2\theta-1}$
Clearly $x$ lies in the III quadrant in which $\tan x$ and $\cot x$ are positive and all the other trigonometric functions are negative.
$\sec x=-2$
$\cos x=\large\frac{1}{\sec x}=-\frac{1}{2}$
$\tan x=\sqrt{\sec^2x-1}$
$\Rightarrow \sqrt{(-2)^2-1}$
$\Rightarrow \sqrt{4-1}$
$\Rightarrow \sqrt 3$
$\tan x=\large\frac{\sin x}{\cos x}$
$\therefore \sin x=\tan x.\cos x$
$\Rightarrow \sqrt 3\times -\large\frac{1}{2}$
$\Rightarrow -\large\frac{\sqrt 3}{2}$
$cosec x=\large\frac{2}{-\sqrt 3}$