Write $\cos(2t) + \sqrt3 \sin(2t)$ in the form $C\cos(w(t-t_0))$ **Problem** Write $\cos(2t) + \sqrt3 \sin(2t)$ in the form $C\cos(w(t-t_0))$ **Attempts** I'm woefully under-practiced in the art of rewriting trigonometric expressions. I've tried looking through the Wikipedia article listing trigonometric identities, but I'm coming up short. Any help and/or solutions appreciated!

Hint use angle sum and difference identities for trignonometric functions: $$C\cos(w(t-t_0))=C(\cos(wt)\cos(wt_0)+\sin(wt)\sin(wt_0))$$ $$=C\cos(wt_0)\cos(wt)+C\sin(wt_0)sin(wt)$$

Compare this expression (for $w=2$) with $$\cos(2t)+\sqrt{3}\sin(2t)$$

After comparing both representations, we can conclude

$$1=C\cos(2t_0)$$ $$\sqrt{3}=C\sin(2t_0)$$

Square both equations and add them to get $$1+3=C^2(\cos^2(2t)+\sin^2(2t))=C^2$$

In the last step we used $\cos^2(2t)+\sin^2(2t)=1$. So $C=\pm2$. Now plug this result into $1=C\cos(2t_0)$ and solve for $t_0$.