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Math Help - Trigonometric equation

  1. #1
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    Trigonometric equation

    Solve the trigonometric equation for the exact values of x, 0<x<2Pi

    2cos(squared) xtanx-tanx=0
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  2. #2
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    2\cos^2 x \tan x - \tan x = 0

    \Rightarrow (\tan x)(2\cos^2 x - 1) = 0

    \Rightarrow (\tan x)(\cos 2x) = 0

    (identity 2\cos^2 x - 1 = \cos 2x)

    To solve the equation, find the solutions for \cos 2x = 0 \,\,\,\mbox{and} \,\,\,\tan x = 0 \,\,\,\,\mbox{in the interval} \,\,\,\,0 \leq x \leq 2\pi
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  3. #3
    A riddle wrapped in an enigma
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    Quote Originally Posted by tradar View Post
    Solve the trigonometric equation for the exact values of x, 0<x<2Pi

    2cos(squared) xtanx-tanx=0
    2\cos^2x \tan x-\tan x=0

    \tan x(2\cos^2x-1)=0

    \tan x=0 \ \ or \ \ 2\cos^x-1=0

    2\cos^2 x=1

    \cos^2 x=\frac{1}{2}

    \cos x=\pm \sqrt{\frac{1}{2}}=\pm \frac{\sqrt{2}}{2}

    \tan x=0 \ \ at \ \ \{0, \pi\}

    \cos x = \pm \frac{\sqrt{2}}{2} \ \ at \ \ \left\{\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\right\}

    Solution set= \left\{0, \pi ,\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\right\}
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