Question

Which of the following is a solution to 3tan^3x=tanx?

Answer 1

Answer:
0, π, 2π, π/6, 5π/6, 7π/6 and 11π/6

Explanation:
3 tan³x = tan x
3 tan³x - tan x = 0
tan x (3 tan²x - 1) = 0
either tan x = 0
This means that:
x = 0 , π or 2π ...........> I

or 3 tan²x - 1 = 0
This means that:
3 tan²x = 1
tan²x = 1/3
tan x = ± $\frac{1}{ \sqrt{3}}$
at tan x = $\frac{1}{ \sqrt{3}}$
x = π/6 or 7π/6 ...................> II
at tan x = - $\frac{1}{ \sqrt{3}}$
x = 5π/6 or 11π/6 ..............> III

From I, II and III, the solutions for x would be:
0, π, 2π, π/6, 5π/6, 7π/6 and 11π/6

Hope this helps :)

Answer 2

The given equation is:

$3 tan^{3}x=tanx \\ \\ 3 tan^{3}x-tanx=0 \\ \\ tanx(3 tan^{2}x-1)=0$

This means:
$tan(x)=0 \\ \\ x= tan^{-1}(0) \\ \\ x= \pi n$
where n is any integer.

&

$3 tan^{2}x-1=0 \\ \\ tan^{2}x = \frac{1}{3} \\ \\ tan(x) = +- \sqrt{ \frac{1}{3} } \\ \\ x= \frac{5 \pi }{6}+n \pi , \frac{ \pi }{6}+n \pi$

The values of x are the solutions to the given trigonometric equations.