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. :)