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Math Help - Trig- question

  1. #1
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    Trig- question

    tan (π/4 + x) = 3tan (π/4 - x) find the values of x.
    Thanks!
    Oh and " π " is pi.
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  2. #2
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    Re: Trig- question

    \tan\left(\frac{\pi}{4} \pm x\right) =\frac{\tan\left(\frac{\pi}{4}\right) \pm \tan{x}}{1 \mp \tan\left(\frac{\pi}{4}\right)\tan{x}} =\frac{1 \pm \tan{x}}{1 \mp \tan{x}}

    so ...

    \tan\left(\frac{\pi}{4} + x\right) = 3\tan\left(\frac{\pi}{4} - x\right)

    \frac{1 + \tan{x}}{1 -\tan{x}} = \frac{3(1 - \tan{x})}{1+\tan{x}}

    (1+\tan{x})^2 = 3(1-\tan{x})^2

    1 + 2\tan{x} + tan^2{x}= 3 - 6\tan{x} + 3\tan^2{x}

    0 = 2\tan^2{x} - 8 \tan{x} + 2

    0 = \tan^2{x} - 4 \tan{x} + 1

    0 = \tan^2{x} - 4 \tan{x} + 4 - 4 + 1

    0 = (\tan{x} - 2)^2 - 3

    \tan{x} = 2 \pm \sqrt{3}

    assuming 0 < x < 2\pi ...

    x = \arctan(2 + \sqrt{3})

    x = \pi + \arctan(2 + \sqrt{3})

    x = \arctan(2 - \sqrt{3})

    x = \pi + \arctan(2 - \sqrt{3})
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