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Math Help - GRADE 12 PRE-CAL IDENTITIES

  1. #1
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    GRADE 12 PRE-CAL IDENTITIES

    question is:

    sin(3x) = 3sinx-4sin³x

    there is a clue..
    it's that sin(3x)=sin (2x+x)

    please help!! and as fast as possible!!
    message me or whatever is possible!
    Last edited by anothergirl; October 29th 2008 at 08:24 PM. Reason: had to add something.
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  2. #2
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    Solution

    sin(3x) = 3sinx-4sin³x

    there is a clue..
    it's that sin(3x)=sin (2x+x)

    We can start by rewriting the left side of the equation as per suggestion:
    sin(2x+x) = 3sinx-4sin³x

    Use the following identity for sine sum formula:
    sin(A+B)=sin A cos B + cos A sin B

    Again, rewrite the left side of the equation with the formula:
    sin2x*cosx + cos2x*sinx = 3sinx - 4sin³x

    Use the following double angle formulas:
    sin2x = 2sinxcosx
    cos2x = (cosx)^2 - (sinx)^2

    Rewrite the left side of the equation using these double angle formulas:
    2sinxcosx*cosx + ((cosx)^2 - (sinx)^2)*sinx = 3sinx - 4sin³x

    Divide both sides by sinx to yield:
    2(cosx)^2 + (cosx)^2 - (sinx)^2 = 3 - 4*(sinx)^2

    Add like terms:
    3*(cosx)^2 = 3 - 3*(sinx)^2

    Add sine term to both sides:
    3*(cosx)^2 + 3*(sinx)^2 = 3

    Divide both sides by 3 and you get the true identity:

    (cosx)^2 + (sinx)^2 = 1

    I hope this is useful.
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