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Math Help - Find all solutions

  1. #1
    Member purplec16's Avatar
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    Find all solutions

    -cos^2x- 4 sin \frac{1}{2}x+2=0
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    Quote Originally Posted by purplec16 View Post
    -cos^2x- 4 sin \frac{1}{2}x+2=0
    note that this is not an easy equation to solve by hand.

    (sure that you posted it correctly?)

    If correct, then I recommend you graph the left side of the equation over the interval from 0 to 2\pi and locate the zeros using your calculator.
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  3. #3
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by purplec16 View Post
    -cos^2x- 4 sin \frac{1}{2}x+2=0
    The sine of the half-angle formula should (?) help here:

    sin(x/2)=+- sqrt[(1−cosx)/2]

    Good luck!
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  4. #4
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    Hello purplec16
    Quote Originally Posted by purplec16 View Post
    -cos^2x- 4 sin \frac{1}{2}x+2=0
    If you're going to try for an analytical solution, you can reduce this equation to a quartic in s, where s = \sin\tfrac12x, using \cos x = 1 - 2\sin^2x. The equation then becomes:
    4s^4 -4s^2+4s-1=0
    but, apart from adopting a numerical/graphical method (as skeeter suggested), I can't see what to do next.

    Grandad
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