pre cal

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what are the amplitude and period of the sinusoid y=6cosx+7sinx?

  • pre cal -

    We could write 6cosx + 7sinx as a single
    sine function of the form asin(x + A)

    asin(x+A) = asinxcosA + acosxsinA = 7sinx + 6cosx
    let x=0
    ---> asinA = 6 or sinA = 6/a
    let x=π/2
    ---> acosA = 7 or cosA = 7/a
    but sin^2 A + cos^2 B = 1
    36/a^2 + 49/a^2 = 1
    a^2 = 85
    a =√85

    so the amplitude is √85
    and the period is 2π, (A is simply a phase shift)

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