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find general solutions for sin5x=cosx

  • maths -

    This is pointless. There are lots of solutions, but they are nasty, and really don't provide any insights.

    sin(5x) = sin^5 + 5sin*cos^4 - 10sin^3cos^2
    = sin*(1-cos^2)^2 + 5sincos^4 - 10sin(1-cos^2)cos^2
    = sin(12cos^4 - 12cos^2 + 1)

    so, you end up trying to solve

    sin(x) (12cos^4(x) - 12cos^2(x) + 1) = cos(x)

    No way to get rid of the sin except to square both sides, and then you wind up with

    144cos^10(x) - 432cos^8(x) + 456cos^6(x) - 192cos^4(x) + 26cos^2(x) - 1 = 0

    So, you have to solve a 5th-degree polynomial in cos^2(x), then try to eliminate possible spurious solutions.

    Good freakin' luck!

    Visit wolframalpha.com to see the solution and the graphs.

    Just type in "solve sin 5x = cos x"

    Not sure what the point is to this exercise.

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