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Math Help - Quick Trig Question

  1. #1
    Super Member
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    Quick Trig Question

    Hi

    Can someone tell me how to do this question:

    1) Convert 4sin2x+5cos2x into a single cosine wave form Acos(2x+C)

    P.S
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  2. #2
    MHF Contributor
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    Quote Originally Posted by Paymemoney View Post
    Hi

    Can someone tell me how to do this question:

    1) Convert 4sin2x+5cos2x into a single cosine wave form Acos(2x+C)

    P.S
    in general ,

    a\sin \theta+b\cos \theta=R\sin(\theta+\alpha)

    where R=\sqrt{a^2+b^2}

    and \tan \alpha=\frac{b}{a}

    It works the same for a\cos \theta+b\sin \theta=R\cos (\theta-\alpha)

    but you flip the sign
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  3. #3
    MHF Contributor
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    Hello Paymemoney
    Quote Originally Posted by Paymemoney View Post
    Hi

    Can someone tell me how to do this question:

    1) Convert 4sin2x+5cos2x into a single cosine wave form Acos(2x+C)

    P.S
    Just to expand a little on mathaddict's reply, use the identity
    \cos(P+Q) = \cos P \cos Q - \sin P \sin Q
    as follows:
    A\cos(2x+C) = 4\sin 2x +5\cos 2x

    \Rightarrow A\cos2x\cos C -A\sin2x\sin C = 4\sin2x+5\cos2x

    \Rightarrow \left\{\begin{array}{l l}A\cos C = 5 & \quad\text{(1)}\\ A\sin C = -4&\quad\text{(2)}\end{array}\right.
    Square (1) and (2) and add:
    A^2 = 5^2+(-4)^2

    \Rightarrow A = \sqrt{41}
    Divide (2) by (1):
    \tan C = -\tfrac45
    Grandad
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