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Thread: Sin and Cos Graph

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    Sin and Cos Graph

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    Quote Originally Posted by Mr_Green View Post
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    The general form will be
    $\displaystyle y = A sin(kx + \phi)$

    Since A is the amplitude, this part is easy: A = 3.

    $\displaystyle y = 3 sin(kx + \phi)$

    Now, we want a 0 of the sine function to be positioned at $\displaystyle x = - \frac{\pi}{2}$. We can do this by setting $\displaystyle kx + \phi = 0$ at $\displaystyle x = - \frac{\pi}{2}$, so we have the equation:
    $\displaystyle k \left ( - \frac{\pi}{2} \right ) + \phi = 0$

    Again, we want a 0 of the sine function to be positioned at $\displaystyle x = \frac{3 \pi}{2}$. We can do this by setting $\displaystyle kx + \phi = \pi$ ($\displaystyle \pi$ is the next zero for the sine function after 0) at $\displaystyle x = \frac{3 \pi}{2}$, so we have the equation:
    $\displaystyle k \left ( \frac{3 \pi}{2} \right ) + \phi = 0$

    Solving the first equation for $\displaystyle \phi$ gives:
    $\displaystyle \phi = \frac{\pi}{2}k$

    Inserting this into the second equation gives:
    $\displaystyle \frac{3\pi}{2}k + \frac{\pi}{2}k = \pi$
    or
    $\displaystyle k = \frac{1}{2}$

    This gives
    $\displaystyle \phi = \frac{\pi}{4}$.

    Thus
    $\displaystyle y = 3 sin \left ( \frac{1}{2}x + \frac{\pi}{4} \right )$.

    -Dan
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    Thanks!
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Mr_Green View Post
    THANKS!



    FOR COS WOULD IT BE:


    3cos(.5x+pi/4)

    ???
    Close. You can do that in two ways. The first is to do it the same way I did below, except to use
    $\displaystyle -\frac{\pi}{2}k + \phi = -\frac{\pi}{2}$ ($\displaystyle -\frac{\pi}{2}$ is a convenient first zero for cos)

    $\displaystyle \frac{3\pi}{2}k + \phi = \frac{\pi}{2}$ (the next zero for cos after $\displaystyle -\frac{\pi}{2}$)

    The other way is to recall that $\displaystyle sin(\theta) = cos(\theta - \pi/2)$ so to switch to cosine we need to "shave" a $\displaystyle \pi/2$ off the argument.

    Either way I got:
    $\displaystyle y = 3 cos \left ( \frac{1}{2}k - \frac{\pi}{4} \right ) $

    -Dan
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