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Thread: Trig Identities

  1. #1
    Junior Member
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    Nov 2009
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    Trig Identities

    Trig Identities-math.jpg

    Can some please explain the steps in-order to show the last statment.

    I have tried using trig identites, however i cant get my head around it.
    Attached Thumbnails Attached Thumbnails Trig Identities-math.gif  
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  2. #2
    Senior Member I-Think's Avatar
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    By letting $\displaystyle x=cos\theta$, differentiating to get $\displaystyle \frac{dx}{d\theta}=-sin\theta$, which becomes $\displaystyle dx=-sin\theta{d\theta}$

    Substituting into the integral, you get

    $\displaystyle \frac{-sin\theta{d\theta}}{(1+cos^2\theta)(\sqrt{1-cos^2\theta})}$

    However $\displaystyle 1-cos^2\theta=sin^2\theta
    $
    So simplifying the expression

    $\displaystyle \frac{-d\theta{sin\theta}}{(1+cos^2\theta)(\sqrt{sin^2\th eta})}$
    $\displaystyle \frac{-d\theta}{1+cos^2\theta}$

    And as we've changed the variable, we need to change the limits of the integral
    As $\displaystyle x=cos\theta$, when$\displaystyle x=1,$ $\displaystyle \theta=0 $
    And when $\displaystyle x=-1, \theta=\pi$

    So the integral becomes
    $\displaystyle 2\int_{\pi}^{0} \frac{-d\theta}{1+cos^2\theta}$

    Becoming
    $\displaystyle -2\int_{\pi}^{0} \frac{d\theta}{1+cos^2\theta}$
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  3. #3
    Junior Member
    Joined
    Nov 2009
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    Trig Identity

    Thanks, that was really helpful, can you please explain:

    Trig Identities-untitled.jpg

    I don't see how the 2 can disappear, and the limits on the integral changes.
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