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

Hint: $f(x)=3\sin(πx)+3\sqrt{3}\cos(πx)$. What is $A\sin(Kx+D)$? It is $A\sin(Kx+D)=A(\sin (Kx) \cos (D) + \cos (Kx) \sin (D))$. Let $A=1$. Then, let $K=\pi$, then $\sin (D)=3\sqrt{3}$, $\cos (D)=3$, i.e., $\tan(D)=\frac{1}{\sqrt{3}}$. Can you find $D$ now?
