Why Does Asin(ax)+Bcos(ax) Create a Sinusoidal Wave?

[mtanti]

Can someone tell me why Asin(ax)+Bcos(ax) always gives another sinusoidal wave?

 

[arildno]

Because you can rewrite that expression into a sinusoidal form.

 

[mtanti]

and can that answer be translated into a more mathematical reason?

 

[neutrino]

Try it out...
put A = C*cos(d) B = C*sin(d) where C and d are arbitrary constants. Of course you can put sin in the place of cos and vice-versa, and still get a sinusoidal wave.

 

[AlphaNumeric]

$$A\sin ax + B \cos ax = R \sin (ax+b)$$

Expanding the right hand side gives you $$R\sin ax \cos b + R \cos ax \sin b$$

This gives $$R \cos b = A$$ and $$R \sin b = B$$, therefore $$b = \tan^{-1}\frac{B}{A}$$.

A similar method gives you R in terms of A and B, thus turning your sum of trig functions into another, single, trig function. :)