Uncertain how to solve this trigonometric equationI 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 kTo 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.

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Uncertain how to solve this trigonometric equationI 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  kTo 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&#039;t agree with the given answers.

enracant60 · Accepted Answer

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 · Accepted Answer

Elaborating on player3236&#039;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.

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

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