Question

Find the general solution of each equation:

$\left(i\right)\sqrt{3}cotx+1=0$

$\left(ii\right)cosecx+\sqrt{2}=0$

Answer 1

$\left(i\right)\sqrt{3}cotx+1=0$

$\Rightarrow cotx=\frac{-1}{\sqrt{3}}$

$\Rightarrow tanx=-\sqrt{3}=-tan\frac{\pi }{3}=tan(\pi -\frac{\pi }{3})=tan\frac{2\pi }{3}$

$\Rightarrow tanx=tan\frac{2\pi }{3}$

$\Rightarrow x=(n\pi +\frac{2\pi }{3})$, where $n\in I.$

Hence, the general solution is $x=(n\pi +\frac{2\pi }{3})$, where $n\in I.$

$\left(ii\right)cosecx+\sqrt{2}=0$

$\Rightarrow sinx=-\frac{1}{\sqrt{2}}=-sin\frac{\pi }{4}=sin(\pi +\frac{\pi }{4})=sin\frac{5\pi }{4}$

$\Rightarrow sinx=sin\frac{5\pi }{4}$

$\Rightarrow x=\{n\pi +(-1{)}^n.\frac{5\pi }{4}\}$, where $n\in I.$

Hence, the general solution is $x=\{n\pi +(-1{)}^n.\frac{5\pi }{4}\}$, where $n\in I$.