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Math Help - evaluating inverse trig functions

  1. #1
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    evaluating inverse trig functions

    evaluate cos(2 tan^-1(1/3))

    I figure if I can evaluate arctan of 1/3 I can solve it (multiply by two and take the cosine of the result).

    Is it advisable to do this with a calculator or is the value of acrtan(1/3) commonly known value?

    thanks!
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    Re: evaluating inverse trig functions

    Hello, kingsolomonsgrave!

    \text{Evaluate: }\:\cos\left[2\tan^{\text{-}1}\!\left(\tfrac{1}{3}\right)\right]

    \text{Let }\theta \:=\:\tan^{\text{-}1}\!\left(\tfrac{1}{3}\right)

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

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


    \text{Therefore: }\:\cos\left[2\tan^{\text{-}1}\!\left(\tfrac{1}{3}\right)\right] \;=\;\cos(2\theta) \;=\;\cos^2\!\theta - \sin^2\!\theta

    . . . . . . . . . . =\;\left(\frac{3}{\sqrt{10}}\right)^2 - \left(\frac{1}{\sqrt{10}}\right)^2 \;=\;\frac{9}{10} - \frac{1}{10} \;=\;\frac{8}{10} \;=\;\boxed{\frac{4}{5}}
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  3. #3
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    Re: evaluating inverse trig functions

    Quote Originally Posted by kingsolomonsgrave View Post
    evaluate cos(2 tan^-1(1/3))

    I figure if I can evaluate arctan of 1/3 I can solve it (multiply by two and take the cosine of the result).

    Is it advisable to do this with a calculator or is the value of acrtan(1/3) commonly known value?

    thanks!
    You don't need to know the value. Remember that inverse trig functions are angles!

    Let \alpha = \tan^{-1}\left( \frac{1}{3}\right)

    So by the double angle identity we have

    \cos(2\alpha)=2\cos^2(\alpha)-1

    Now we just need to figure out

    \cos(\tan\left( \frac{1}{3}\right)

    So we can draw a triangle and use the pythagorean theorem

    evaluating inverse trig functions-capture.png

    This shows that

    \cos(\tan\left( \frac{1}{3}\right)=\frac{3}{\sqrt{10}}


    Now just plug this in to finish..
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