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Thread: please show steps

  1. #1
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    please show steps

    solve:
    cos^2(x) + cos(x) = cos2x
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by chopstixdcb View Post
    solve:
    cos^2(x) + cos(x) = cos2x
    Note that $\displaystyle \cos(2x)=2\cos^2x-1$.

    Therefore, your equation becomes $\displaystyle \cos^2x-\cos x-1=0$.

    Letting $\displaystyle u=\cos x$, we end up with the quadratic equation $\displaystyle u^2-u-1=0$.

    Can you take it from here?
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  3. #3
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    so now, all i have to do is factor it right?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by chopstixdcb View Post
    so now, all i have to do is factor it right?
    If possible; Otherwise, resort to quadratic formula.

    Also, remember this is a quadratic in terms of $\displaystyle \cos x$, so once you have your solutions, you need to solve $\displaystyle \cos x=u_1$ and $\displaystyle \cos x=u_2$ to find the solutions that you're seeking.
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  5. #5
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    oh geez...i feel so dumb right now. okay thanks a lot.
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  6. #6
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    Quote Originally Posted by Chris L T521 View Post
    If possible; Otherwise, resort to quadratic formula.

    Also, remember this is a quadratic in terms of $\displaystyle \cos x$, so once you have your solutions, you need to solve $\displaystyle \cos x=u_1$ and $\displaystyle \cos x=u_2$ to find the solutions that you're seeking.
    umm, if you don't mind, can you show me the final steps. for some reason i got non real solution and the other one is cosx=1
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  7. #7
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    $\displaystyle u^2 - u - 1 = 0$

    $\displaystyle u^2 - u + \left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right)^2 - 1 = 0$

    $\displaystyle \left(u - \frac{1}{2}\right)^2 - \frac{5}{4} = 0$

    $\displaystyle \left(u - \frac{1}{2}\right)^2 = \frac{5}{4}$

    $\displaystyle u - \frac{1}{2} = \frac{\pm \sqrt{5}}{2}$

    $\displaystyle u = \frac{1 \pm \sqrt{5}}{2}$.


    Note that $\displaystyle -1 \leq \cos{x} \leq 1$ for all $\displaystyle x$. Are there any solutions that don't fit?
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