Question

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

7tan^3x-21tanx=0
a. pi/3, 2pi/3, 4pi/3, 5pi/3
b. 0, pi/5, pi, 6pi/5
c. 0, pi/3, 2pi/3, pi, 4pi/3, 5pi/3
d. 0, pi/3, pi, 4pi/3

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{\frac{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 + \frac{\pi}3\\\\\implies x = \frac{\pi}3, ~~(4 \pi)/(3) ~~~~~~~~;[\text{For n = 0,1 and }~ [0, 2\pi)]\\\\\\`

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

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