Rewrite $f(x) = 3 \sin (\pi x) + 3\sqrt{3} \cos (\pi x)$ in the form $f(x) = A \sin (Kx+D)$ I got a question like that said "Rewrite $f(x) = 3 \sin (\pi x) + 3\sqrt{3} \cos (\pi x)$ in the form $f(x) = A \sin (Kx+D)$".
I'm inclined to think that since the periods are the same ($2$), that the amplitudes will just add up. But, I'm not sure.
I also need to know the rules for combining sinusoids with different periods. I know that when you're multiplying them, the one with the longer period acts as a sort of envelope for the one with the smaller period. But what do you do with different periods when you add them?
Thanks!
evamvid

The standard way to combine such functions is to use the R Formula ( http://www.oocities.org/maths9233/Trigonometry/RFormula.html) :
Specifically, if we wanna combine your function into : 
$$a\sin{\theta} + b\cos{\theta} = R\sin{(\theta + \alpha)}$$
Then it is possible to compute $R,\alpha$ : 
$$R = \sqrt{a^2 + b^2}, \alpha=\tan^{-1}{\frac{b}{a}}$$
In your case, $$ a=3, b = 3\sqrt{3}, \theta=\pi x$$
I believe it's very easy to continue on from here. Im sure there are many proofs of this online as well, if you need to be convinced that it works.
