Is there a trigonometric identity for this ?

[Usjes]

Hi,

I am trying to figure out what the result is when adding two sinusoids of the same frequency but with different phase and amplitudes. Specifically I want to know if the result is always another sinusoid of the same frequency. For the case of the the same amplitude I have:

cos(wt) + cos(wt+phi) = 2.cos(wt+phi/2).cos(phi/2) ... this follows from the trig. identity:

cos(A)+cos(B) = 2.cos[(A+B)/2].cos[(A-B)/2]

So the addition of two sinusoids of the same frequency and amplitude will always give a sinusoid of the same frequency. What about the more general case:
cos(wt) + k.cos(wt+phi) where k is a constant. Will the result here also always be a sinusoid of frequency w ? Intuitively I think so, but I havn't been able to derive an expression for the phase and amplitude.
Does anyone know ?

Thanks.

 

[wnvl]

cos(wt) + k.cos(wt+phi)=-k sin(phi) sin(wt)+(k cos(phi)+1)cos(wt)

 

[AlephZero]

cos(wt) + k.cos(wt+phi)=-k sin(phi) sin(wt)+(k cos(phi)+1)cos(wt)

... and if you want, you can write any expression of the form
A sin(wt) + B cos(wt)
in the form
sqrt(A^2 + B^2)cos(wt + alpha)
where
tan alpha = A/B