How can I simplify adding two sine waves using trigonometric identities?

[flexifirm]

Add the following sine waves

y= Asin(wt) + Bsin(wt + x) Where w=2(pi)f and x is a phase shift


I SIMPLIFIED this for my friend using rotating vectors, then i got really annoyed realizing that I didn't know how to do it the direct way (trig identities).

So I've posted it here for all you smart folks to try

 

[matt grime]

What do you mean by solved?

 

[matt grime]

Ok, after consideration, is this what you want?

We may use sin(u+v)=sinucosv+cosusinv to rewrite sin(wt+x) in terms of cos(wt) and sin(wt), and then collecting terms we can rewrite the equation as Csinwt+Dcoswt for some choice of C and D. You may then blindly apply a formula whose derivation comes from :

suppose Csinz+Dcosz= Rsin(z+q), then C = Rcos(q) and D=Rsin(q) after applying that identity, from which we see R=sqrt(C^2+d^2) and tan(q) = D/C ie q=arctan(D/C)

So, you may conclude that y= Rsin(wt+q) for suitable R and q which can be derived with a little work. You can use that to do most anything you wish now.