Question

Find all solutions in the interval [0,2pi).

Answer 1

$7 \tan^3 x - 21\tan x =0\\\\\implies 7\tan x(7 \tan^2 x -21) = 0\\\\\implies 7 \tan x = 0 ~~\text{or}~~ 7\tan^2 x -21 =0\\\\\implies \tan x = 0 ~~\text{or}~~ \tan x = \pm\sqrt{\dfrac{21}7} = \pm \sqrt 3\\\\\text{Now,}\\\\\tan x = 0\\\\\implies x = n \pi \\\\\implies x = 0, \pi ~~~~~~~~~~~~~;[\text{For n=0,1 and}~ [0, 2 \pi)}]\\\\\\\tan x = \sqrt 3 \\\\\implies x = n \pi + \dfrac{\pi}3\\\\\implies x = \dfrac{\pi}3, ~~\dfrac{4 \pi}{3} ~~~~~~~~;[\text{For n = 0,1 and }~ [0, 2\pi)]$

$\tan x = -\sqrt 3\\\\\implies x= n\pi - \dfrac{\pi}3\\\\\implies x = \dfrac{2\pi}{3},~~ \dfrac{5\pi}3 ~~~~~~~ ;[\text{For n=1,2 and}~ [0,2\pi)]$'

$\text{Combine all solutions,}\\\\x= 0, ~\pi, ~\dfrac{\pi}3,~ \dfrac{4 \pi}3 ,~ \dfrac{2 \pi}3 , ~\dfrac{5\pi}3$