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Math Help - [SOLVED] Trig Identities

  1. #1
    Junior Member madmax29's Avatar
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    Exclamation [SOLVED] Trig Identities

    Hi there

    Ive been trying out this identity;
    Prove:

    Sinē Θ =1/2 (1- Cos 2Θ)

    I got to
    Sinē Θ = 1-Cos 2Θ/ 2

    but i am unsure about what to do to get out of the next problem, can you cancel the 2???


    Could you show how you continue?

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  2. #2
    Super Member

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    Hello, madmax29!

    This is one of the basic trig identities.

    What facts and identities are we allowed to use?


    Prove: . \sin^2\!\theta \:=\:\frac{1}{2}(1-\cos2\theta)

    The right side is: . \frac{1-\cos2\theta}{2} .[1]


    If we are allowed the identity: . \cos(A+B) \:=\:\cos A\cos B - \sin A\sin B

    . . then we have: . \cos(\theta + \theta) \:=\:\cos\theta\cos\theta - \sin\theta\sin\theta \quad\Rightarrow\quad\cos2\theta \;=\;\cos^2\!\theta - \sin^2\!\theta

    Since \cos^2\!\theta \:=\:1-\sin^2\!\theta, we have: . \cos2\theta \:=\:(1-\sin^2\!\theta) - \sin^2\!\theta \quad\Rightarrow\quad\cos2\theta \:=\:1 - 2\sin^2\!\theta


    Substitute into [1]: . \frac{1 - (1-2\sin^2\!\theta)}{2}\;=\;\frac{2\sin^2\!\theta}{2} \;=\;\sin^2\!\theta

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