Question

Find the principal and general solutions of the following equations:(i) $\tan x=\sqrt{3}$ (ii) $\sec x=2$(iii) $\cot x=-\sqrt{3}$(iv) $cosecx=-2$

Answer 1

(i) $\tan x=\sqrt{3}=\tan \frac{\pi }{3}=\tan \frac{4\pi }{3}$
Hence principal solution of $\tan x=\sqrt{3}$
are $\frac{\pi }{3}$ and $\frac{4\pi }{3}$

$\tan x=\sqrt{3}=\tan \frac{\pi }{3}$

Hence general solution is given by,

$x=n\pi +\frac{\pi }{3}$ where $n\in Z$

(ii) $\sec x=2=\sec \frac{\pi }{3}=\sec \frac{5\pi }{3}$

Hence principal solution of $\sec x=2$

are $\frac{\pi }{3}$ and $\frac{5\pi }{3}$

$\sec x=2=\sec \frac{\pi }{3}$

Hence general solution is given by,

$x=2n\pi \pm \frac{\pi }{3}$, where $n\in Z$

(iii)$\cot x=-\sqrt{3}=\cot \frac{5\pi }{6}=\cot \frac{11\pi }{6}$

Hence principal value of $\cot x=-\sqrt{3}$

are $\frac{5\pi }{6}$ and $\frac{11\pi }{6}$

$\cot x=-\sqrt{3}=\cot \frac{5\pi }{6}$

Hence general solution is given by,

$x=n\pi +\frac{5\pi }{6}$, where $n\in Z$

(iv) $\text{csc}x=-2=\text{csc}\frac{7\pi }{6}=\text{csc}\frac{11\pi }{6}$

Hence principal solution of $\text{csc}x=-2$

are $\frac{7\pi }{6}$ and $\frac{11\pi }{6}$

$\text{csc}x=-2=\text{csc}\frac{7\pi }{6}$

Hence general solution is given by,

$x=n\pi +(-1{)}^n\frac{7\pi }{6}$, where $n\in Z$