simplification of trigonometric equation

• April 20th 2012, 01:38 AM
niaren
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
• April 20th 2012, 01:52 AM
princeps
Re: simplification of trigonometric equation
Hint :

$\sin^2 w=\frac{\tan^2 w}{1+\tan^2 w} ~\text{and}~ \cos^2 w=\frac{1}{1+\tan^2 w}$
• April 20th 2012, 02:42 AM
niaren
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.