Trigonometric equality involving $a\sin \alpha + b \sin \beta$

Question

I was watching this video on orthogonal signals and the author used a trigonometric equality that I haven't seen before (at the bottom of the screenshot):

Could anyone tell me where I can find a proof of this?

Answer 1

As in the image, $\sin\left(\theta+\dfrac\pi2\right)=\cos\theta$

Now $A\sin(wt+\psi)+B\sin(wt+\phi)=\sin wt(A\cos\psi+B\cos\phi)+\cos wt(A\sin\psi+B\sin\phi)$

Now can you derive $p\sin x+q\cos x =\sqrt{p^2+q^2}\sin\left(x+\arcsin\dfrac p{\sqrt{p^2+q^2}}\right)$