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Math Help - simplification of trigonometric equation

  1. #1
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    simplification of trigonometric equation

    I have an expression on the form
    Code:
    a*sin(w)3 + b*sin(w)
    --------------------    (1)
    c*cos(w)3 + d*cos(w)
    
    
    My goal is to arrive at an expression on this form


    Code:
       
       sin(w)3       sin(w)
    p*--------- + q*-------    (2)
       cos(w)3       cos(w)
    
    
    I believe it should be possible, I just can't find a way to manipulate (1) in to (2).

    Any help is much appreciated!

    Br
    niaren
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  2. #2
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    Crna Gora
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    Re: simplification of trigonometric equation

    Hint :

    \sin^2 w=\frac{\tan^2 w}{1+\tan^2 w} ~\text{and}~ \cos^2 w=\frac{1}{1+\tan^2 w}
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  3. #3
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    Re: simplification of trigonometric equation

    Holy smoke, that was fast!

    I got it now.
    I'm a little surprised you came up with the hint so fast.
    For the equality to hold, d == 0 must be satisfied.

    In fact, I'm considering a more general problem of manipulating

    Code:
    aN*sin(w)^N + .... + a3*sin(w)^3 + a1*sin(w)
    ------------------------------------------------   (1)
    bN*cos(w)^N + .....+ b3*cos(w)^3 + b1*cos(w)
    into

    Code:
          
        sin(w)^N       sin(w)^3         sin(w)
    cN*--------- + c3 ----------- + c1*--------    (2)
        cos(w)^N       cos(w)^3         cos(w)
    
    Where N is odd.

    I suspect your hint can be used for any N by using appropriate powers of you hints. I'll look into that.
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