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Thread: Harmonic Identities Help

  1. #1
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    Harmonic Identities Help

    The third Harmonic of a sound is given by 4 cos x -6 sin

    Using the Harmonic Identity A cos x B sin ≡ R cos (x + α)
    R is given in Cos and alpha.

    Yet I need to answer in the form R sin (3 θ + β)
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  2. #2
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    Re: Harmonic Identities Help

    One thing you need is that $\displaystyle cos(p+ q)= cos(p)cos(q)- sin(p)sin(q)$. From that, Rcos(x+ a)= Rcos(a)cos(x)- Rsin(a)sin(x). So you want $\displaystyle \alpha$ and R such that $\displaystyle R cos(\alpha)= 4$ and $\displaystyle R sin(\alpha)= 6$. Then $\displaystyle R^2cos^2(\alpha)+ R^2sin^2(\alpha)= R^2= 4^2+ (-6)^2= 16+ 36= 52$. $\displaystyle R= \sqrt{52}= 2\sqrt{13}$. Then $\displaystyle R cos(\alpha)= 2\sqrt{13} cos(\alpha)= 4$. $\displaystyle cos(\alpha)= \frac{2}{\sqrt{13}}= \frac{2\sqrt{13}}{13}$ so $\displaystyle \alpha= cos^{-1}\left(\frac{2\sqrt{13}}{13}\right)$.
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    Re: Harmonic Identities Help

    I,m Really sorry HallsofIvy I have wrote the equation incorrectly

    The third Harmonic of a sound is given by 4 cos (3 θ) - 6 sin (3 θ), Using the Harmonic Identity A cos x – B sin ≡ R cos (x + α)

    R is given in Cos and alpha.

    Yet I need to express this sound wave in the form R sin (3 θ + β)
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  4. #4
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    Re: Harmonic Identities Help

    Quote Originally Posted by andyt View Post
    I,m Really sorry HallsofIvy I have wrote the equation incorrectly

    The third Harmonic of a sound is given by 4 cos (3 θ) - 6 sin (3 θ), Using the Harmonic Identity A cos x B sin ≡ R cos (x + α)

    R is given in Cos and alpha.

    Yet I need to express this sound wave in the form R sin (3 θ + β)
    $R \sin(3\theta+\beta) = R \sin(3\theta)\cos(\beta) + R\cos(3\theta)\sin(\beta)$

    equating like terms we get

    $R\cos(\beta) = -6$

    $R\sin(\beta) = 4$

    $\tan(\beta) = -\dfrac 2 3$

    $\beta = \arctan\left(-\dfrac 2 3\right)$

    $\beta = \pi -\tan ^{-1}\left(\dfrac{2}{3}\right)$

    $R^2 = (-6)^2 + (4)^2 = 52$

    $R = \sqrt{52}=2\sqrt{13}$


    $4 cos (3 \theta) - 6 sin (3 \theta) = 2\sqrt{13}\sin\left(3\theta + \pi -\tan ^{-1}\left(\dfrac{2}{3}\right)\right)$
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    Re: Harmonic Identities Help

    Morning Romsek

    Rsin(3θ+β)≡Acos(3θ)-Bsin(3θ)

    What equation did you use to get to: R cos (β) = -6 and R sin (β) = 4



    Thanks
    Andy
    Last edited by andyt; Jan 25th 2019 at 01:09 AM. Reason: correction
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  6. #6
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    Re: Harmonic Identities Help

    Quote Originally Posted by andyt View Post
    Morning Romsek

    Rsin(3θ+β)≡Acos(3θ)-Bsin(3θ)

    What equation did you use to get to: R cos (β) = -6 and R sin (β) = 4



    Thanks
    Andy
    I equated coefficients of $\cos(3\theta)$ and $\sin(3\theta)$
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    Re: Harmonic Identities Help

    Thanks romsek
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