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Thread: 12cos(ot) + 5sin(ot) + 3cos(ot − pi/3) in the form Acos(ot + phi)

  1. #1
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    12cos(ot) + 5sin(ot) + 3cos(ot − pi/3) in the form Acos(ot + phi)

    Write f(t) = 12\cos(\omega t) + 5\sin(\omega t) + 3\cos(\omega t - \pi/3) in the form f(t) = A\cos(\omega t + \phi).

    My attempt:

    \\ f(t) = 12\cos(\omega t) + 5\sin(\omega t) + 3\cos(\omega t - \pi/3)\\ = 12\cos(\omega t) + 5\sin(\omega t) + 3\cos(\omega t)(cos(-\frac{\pi}{3}) - 3\sin(\omega t)\sin(-\frac{\pi}{3}) \\ = 12\cos(\omega t) + 5\sin(\omega t) + \frac{3}{2} \cos(\omega t) + \frac{3\sqrt{3}}{2}\sin(\omega t) \\ = \frac{27}{2}\cos(\omega t) + \frac{(10 + 3\sqrt{3})}{2}\sin(\omega t).

    I don't know if there is anywhere that I can go from here!

    Any help would be awesome.
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  2. #2
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    Re: 12cos(ot) + 5sin(ot) + 3cos(ot − pi/3) in the form Acos(ot + phi)

    $12\cos(\omega t)+5\sin(\omega t) +3\bigg[\cos(\omega t)\cos\left(\dfrac{\pi}{3}\right)+\sin(\omega t)\sin\left(\dfrac{\pi}{3}\right) \bigg]$

    $\dfrac{27}{2}\cos(\omega t) + \left(\dfrac{10+3\sqrt{3}}{2}\right) \sin(\omega t)$

    note ...

    $A\cos{x}+B\sin{x} = R\cos(x-\alpha)$ where $R= \sqrt{A^2+B^2}$ and $\tan{\alpha} = \dfrac{B}{A}$

    see what you can do from here ...
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    Re: 12cos(ot) + 5sin(ot) + 3cos(ot − pi/3) in the form Acos(ot + phi)

    Quote Originally Posted by skeeter View Post

    $A\cos{x}+B\sin{x} = R\cos(x-\alpha)$ where $R= \sqrt{A^2+B^2}$ and $\tan{\alpha} = \dfrac{B}{A}$

    see what you can do from here ...
    Thanks a lot for the response.

    I did actually use that method and got 15.49 \cos(\omega t - 0.51) which seems to be about right.

    The context of the question seems to suggest that there is a way to get an exact expression without decimals. Is there a way to do this (perhaps involving exponentials?)?

    Thanks again!
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  4. #4
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    Re: 12cos(ot) + 5sin(ot) + 3cos(ot − pi/3) in the form Acos(ot + phi)

    best I can get for an "exact" expression ...

    $R \cos(\omega t - \alpha)$ where $R = \sqrt{15\sqrt{3}+214}$ and $\alpha = \arctan\left(\dfrac{10+3\sqrt{3}}{27}\right)$

    ... maybe someone else will come along and offer an improved version.
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