If we plot the curve

it looks like so:

This looks suspiciously like a transformed $\sin$ curve! It's just been translated and enlarged. So, maybe we should hope that there exist some $R$ and $\alpha$ such that

The $R$ here is what provides the stretch, and the $\alpha$ provides the translation.

In this tutorial, we'll learn how to find $R$ and $\alpha$ so that we can simplify expressions like $2\sin (x)+\cos (x)$ into a single trig function.

1. Express

   $$2\sin (x)+\cos (x)$$

   in the form

   $$R\sin (x+\alpha )$$

   where $R>0$ and $0<\alpha <\pi .$

2. Express

   $$2\sqrt{2}\cos (x)-4\sqrt{2}\sin (x)$$

   in the form

   $$R\cos (x+\alpha )$$

   where $R>0$ and $0<\alpha <\pi .$

3. Hence, solve the equation

   $$\sqrt{30}\cos x-\sqrt{15}\sin x=\sqrt{5},0<x<2\pi$$