How to prove the trigonometric formula $a \sinθ + b \cosθ = R \sin(θ + α)$?

Question

I recently learnt the formula $a \sinθ + b \cosθ = R \sin(θ + α)$ at school. How to prove it? Thanks.

Answer 1

HINT Suffices to show there exist $a,b,R$ such that $$a\sin t + b\cos t = r\sin(t+z).$$

Note that $$ \sin(t+z) = \sin t \cos z + \cos t \sin z $$ and pick $a/r = \cos z, b/r = \sin z$ where $a^2+b^2 = r^2$.

Answer 2

Let $\dfrac{b}{a}=\tan\alpha$ ($a\neq0$) then $$LHS=a\left(\sin\theta+\dfrac{b}{a}\cos\theta\right)=a\left(\sin\theta+\tan\alpha\cos\theta\right)=\dfrac{a}{\cos\alpha}\left(\sin\theta\cos\alpha+\sin\alpha\cos\theta\right)=\dfrac{a}{\cos\alpha}\sin(\theta+\alpha)$$