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Trigonometric equation with tangent

  1. Jul 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Solve: 2 tan x (tan x - 1) = 3.


    2. Relevant equations

    Pythagorean identities?

    3. The attempt at a solution

    I tried the following:

    2 tan^2 x - 2 tan x = 3
    2 (sec^2 x - 1) - 2 tan x = 3
    2 (1 - cos^2 x) - 2 sin x cos x = 3 cos^2 x (multiplying through by cos^2 x)

    ...but then I got confused (not wanting to write sin x = sqrt(1 - cos^2 x) and end up with an algebraic mess). Is there an easier way to solve the problem? Thanks!
     
  2. jcsd
  3. Jul 27, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Use the quadratic equation. Substitute u=tan(x). Now your first equation is 2u^2-2u=3. Solve that for u. Then find x.
     
  4. Jul 28, 2013 #3

    interhacker

    User Avatar
    Gold Member

    Solve the equation as a quadratic, initially.

    [itex]

    y = \tan x \\

    \Rightarrow 2y(y-1) = 3 \\
    \Rightarrow 2y^2 - 2y - 3 = 0 \\

    \Rightarrow y = \frac {2 \pm \sqrt {28}}{4} \\
    \Rightarrow y = \frac {2 \pm 2\sqrt{7}}{4} \\
    \Rightarrow y = \frac {1 \pm \sqrt{7}}{2}\\

    [/itex]

    Then deal with the trig ratio.

    [itex]

    \tan x = \frac {1 \pm \sqrt{7}}{2} \\
    \Rightarrow x = \arctan \frac {1 + \sqrt{7}}{2} \| x = \arctan \frac {1 - \sqrt{7}}{2}\\

    Between\, the\, interval:\\ 0 < x < 2\Pi\\
    x = 1.07\, rad., 4.20\, rad., 2.45\, rad., 5.59\, rad.
    [/itex]
     
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