Uncertain how to solve this trigonometric equation I am currently attempting to discover how to find the general solutions to sqrt 3 tan^2x=2 tan x+ sqrt 3

caschaillo7

caschaillo7

Answered question

2022-10-29

Uncertain how to solve this trigonometric equation
I am currently attempting to discover how to find the general solutions to
3 tan 2 x = 2 tan x + 3
The given solutions are x = π 3 + π k , 5 π 6 + π k
To solve this equation I removed the square root from 3 tan 2 x leaving me with tan x. Then I subtracted 2 tan x from 3 tan x leaving me with tan x = 3
Should I then not solve tan x = 3 for the generalized solutions?
This would give x = π 3 and x = 2 π 3 which would then be generalized to π 3 + π k and 2 π 3 + π k, which doesn't agree with the given answers.

Answer & Explanation

enracant60

enracant60

Beginner2022-10-30Added 10 answers

Let  u = tan x .
Divide through by 3 .
u 2 2 3 u 1 = 0.
This is a quadratic equation, the solutions are:
u = 1 3 2 3 .
We thus have
tan x = 3  or  tan x = 1 3 .
So
x { π 3 + k π , 5 π 6 + k π } .
Alisa Taylor

Alisa Taylor

Beginner2022-10-31Added 4 answers

Elaborating on player3236's comment: Let u = tan x. Then your equation is 3 u 2 = 2 u + 3 . Solve this quadratic equation, and then use u = tan x to solve for x.

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