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Math Help - Find sin(4x) if tan(x) = 4

  1. #1
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    Find sin(4x) if tan(x) = 4

    if find
    Last edited by mr fantastic; January 10th 2010 at 01:09 PM. Reason: Changed post title
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  2. #2
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    I can solve it in two steps,

    First find using the identity,



    so = 2(4)/(1-16) = -8/15

    Now find using,



    Hence,


    =
    =????
    = -
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  3. #3
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    Thank you very much
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  4. #4
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    Hello, dapore!

    bandedkrait is absolutely correct!
    Here's my approach . . .


    If \tan x = 4, find \sin4x
    We have: . \tan x \:=\:\frac{1}{4} \:=\:\frac{opp}{adj}

    Then: .  opp = 1,\;adj = 4\quad\Rightarrow\quad hyp = \sqrt{17}

    . . Hence: . \sin x = \frac{4}{\sqrt{17}},\;\cos x \,=\,\frac{1}{\sqrt{17}}


    And we have: . \begin{array}{ccccccc}\sin2x &=& 2\sin x\cos x &=&2\left(\frac{4}{\sqrt{17}}\right)\left(\frac{1}  {\sqrt{17}}\right) &=& \dfrac{8}{17} \\ \\[-3mm]<br />
\cos2x &=& 1-2\sin^2\!x &=& 1 - 2\left(\frac{4}{\sqrt{17}}\right)^2 &=& \text{-}\dfrac{15}{17} \end{array}


    Therefore: . \sin4x\;=\;2\sin2x\cos2x \;=\;2\left(\frac{8}{17}\right)\left(-\frac{15}{17}\right) \;=\;-\frac{240}{289}

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