Given that $\displaystyle \alpha$ is acute and that $\displaystyle tan \alpha = \frac {3}{4}$, prove that:

$\displaystyle 3 sin (\theta + \alpha) + 4 cos (\theta + \alpha) = 5 cos \theta$

I start by saying that $\displaystyle R cos \alpha = 4 $ and $\displaystyle R sin \alpha = 3 $ but then I get stuck.