Question

Find the principal and general solutions of the following equation.
cosec x = - 2

Answer 1

Here, cosecx= - 2 $\Rightarrow$ sin x =- $\frac{1}{2}$, which is negative, so x lies in third or fourth quadrant.

$\therefore$ sin x =- $\frac{1}{2}$ =- sin ${30}^0$

= sin $({180}^0+{30}^0)$ or sin $({360}^0-{30}^0)$

= sin ${210}^0$ or sin ${330}^0$

= sin $\frac{7\pi }{6}$ or sin $\frac{11\pi }{6}$

Hence the principal solutions are $\frac{7\pi }{6},\frac{11\pi }{6}$

Now sin x =- sin $\frac{\pi }{6}$

$\Rightarrow x=n\pi +(-1{)}^n\left(\frac{7\pi }{6}\right)$ where n $ϵ$ Z.