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Math Help - how does one evaluate cos (2*arctan of (1/3))?

  1. #1
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    how does one evaluate cos (2*arctan of (1/3))?

    how does one evaluate cos (2*arctan of (1/3))?-screen-shot-2012-10-24-1.35.23-am.png

    I can do this on my calculator quite easily but I'm not sure how to do it algebratically
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  2. #2
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    Re: how does one evaluate cos (2*arctan of (1/3))?

    Hello, kingsolomonsgrave!

    \text{Evaluate: }\:\cos\left(\arctan\tfrac{1}{3}\right)

    \text{Let }\,\theta \,=\,\arctan\tfrac{1}{3}

    \text{Then: }\:\tan\theta \:=\:\tfrac{1}{3} \:=\:\tfrac{opp}{adj}

    \theta\text{ is in a right triangle with: }opp = 1,\;adj = 3
    . . \text{Pythagorus says: }\:hyp = \sqrt{10}
    \text{Hence: }\:\cos\theta \:=\:\tfrac{3}{\sqrt{10}}


    \text{Therefore: }\:\cos\left(\arctan\tfrac{1}{3}\right) \;=\;\cos\theta \;=\;\frac{3}{\sqrt{10}}
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    Re: how does one evaluate cos (2*arctan of (1/3))?

    thanks!

    So in order to work with 2*acrtan, at which stage would I multiply by 2?


    would I multiply tan theta by 2 and thus take the cosine of 2/3?

    I may need sleep soon, I think my brain is shutting down, this seems like it should be obvious. Sorry if im being dense
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: how does one evaluate cos (2*arctan of (1/3))?

    Use a double-angle identity for cosine.
    Thanks from Siron
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