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Math Help - simplification of a cos

  1. #1
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    simplification of a cos

    can anybody show me how to simplify: cos(arctan x), im not too sure on how to evaluate that, thanks!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ah-bee View Post
    can anybody show me how to simplify: cos(arctan x), im not too sure on how to evaluate that, thanks!
    Try to construct a triangle with an angle represented by \vartheta=\tan^{-1}x

    This implies that our triangle has an opposite of length x and an adjacent of length 1. We can now use this to determine the length of the hypotenuse of this triangle.

    Using Pythagoras' Theorem, we see that x^2+1^2=h^2\implies h=\sqrt{x^2+1}

    Thus, \cos\vartheta=\frac{adj.}{hyp.}

    Since adj.=1 and hyp.=\sqrt{x^2+1}, we see that \cos(\tan^{-1}x)=\cos\vartheta=\color{red}\boxed{\frac{1}{\sqr  t{x^2+1}}}

    Does this make sense?

    --Chris
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  3. #3
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    omg the way u put it makes it look so easy. very easily understandable!
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