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Thread: change to the form Acos(x+w)

  1. #1
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    change to the form Acos(x+w)

    How do you change F(x) = -cosx + sqrt3sint to the form:

    Acos(x+w)

    with A and w both exact values?
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  2. #2
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    Hello JQ2009
    Quote Originally Posted by JQ2009 View Post
    How do you change F(x) = -cosx + sqrt3sint to the form:

    Acos(x+w)

    with A and w both exact values?
    I assume you mean
    $\displaystyle F(x)= -\cos x + \sqrt3\sin x$
    So, let $\displaystyle F(x) = A\cos(x+w)$. Then:
    $\displaystyle -\cos x + \sqrt3\sin x=A\cos(x+w)$
    $\displaystyle =A\cos x \cos w - A\sin x\sin w$
    Compare coefficients of $\displaystyle \cos x$ and $\displaystyle \sin x$:
    $\displaystyle -1 = A\cos w$

    $\displaystyle \sqrt3 = -A\sin w$
    Square and add:
    $\displaystyle 4 = A^2(\cos^2w+\sin^2w)$

    $\displaystyle \Rightarrow A = -2$ (taking - sign)
    Divide second equation by the first:
    $\displaystyle \tan w = \sqrt3$

    $\displaystyle \Rightarrow w = \pi/3$

    $\displaystyle \Rightarrow F(x) = -2\cos(x+\pi/3)$
    Grandad
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