Summing cosines of different amplitude

[radiogaga35]

Hi there

I am trying to sum many cosines of different amplitude and phase shift, but same ang. frequency (it's not a coursework question). My first thoughts are to sum them two at a time (to simplify matters?), probably using complex numbers. I tried doing it symbolically in MATLAB but it wasn't able to simplify things. Supposing the ang. frequency is 1, I know that the solution can be written:

$$\displaystyle{<br /> C\cos (t + \delta ) + D\cos (t + \varepsilon ) = E\cos (t + \varphi)}$$

where I would have to solve for E and phi. Or equivalently:

$$\displaystyle{Ae^{i(t + \delta )} + e^{i(t + \varepsilon )} = Be^{i(t + \varphi )}}$$

where I would have to solve for B and phi.

Then I split things into two equations (one using real part/cosines, other using imag. part/sines), and eliminate B. Unfortunately this approach doesn't seem to help, as I just end up with a messy arctan of sums of sines and cosines (of different amplitudes -- i.e. back to original problem!).

Any suggestions as to a more fruitful approach?

Thank you.

 

[radiogaga35]

Ok, nevermind, got it figured out! Just used a bit of geometry