SOLUTION: Solve the equation on the interval [0,2pi] sqrt(3) tan^2 x = 3 tan x

Algebra ->  Equations -> SOLUTION: Solve the equation on the interval [0,2pi] sqrt(3) tan^2 x = 3 tan x       Log On


   

Question 1121660: Solve the equation on the interval [0,2pi]
sqrt(3) tan^2 x = 3 tan x
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+sqrt%283%29%2Atan%5E2%28+x+%29+=+3%2Atan%28x%29+
+sqrt%283%29%2Atan%28x%29%2A%28+tan%28x%29+-+sqrt%283%29+%29+=+0+
This is true if either:
+tan%28x%29+=+0+
And +x+=+0+, +x+=+2pi+
Also
+tan%28x%29+=+sqrt%283%29+
+x+=+pi%2F3+

Answer by ikleyn(52864) About Me  (Show Source):
You can put this solution on YOUR website!
.

The solution by @josmiceli was partly incomplete and partly incorrect - so I put my solution below. 


sqrt%283%29 tan^2  x = 3tanx  ====>   divide both sides by sqrt%283%29  ====>


tan^2(x) = sqrt%283%29%2Atan%28x%29


tan%5E2%28x%29+-+sqrt%283%29%2Atan%28x%29 = 0


tan%28x%29%2A%28tan%28x%29+-+sqrt%283%29%29 = 0


This equation deploys in two separate equations:


    1)  tan(x) = 0,  which has two solutions in the given interval:  x= 0  and  x= pi;


and


    2)  tan(x) = sqrt%283%29,  which also has two solutions in the given interval  x= pi%2F3  and  x= 4pi%2F3.


Answer.  The original equation has 4 solutions in the given interval:


                0,  pi,  pi%2F3  and  4pi%2F3.

Solved.