Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$ 
$$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$

I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$
enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$
But I can find how to continue from here

Take two vectors: $v_1 = (a,b)$ and $v_2 = (\sin x, \cos x)$. Their scalar product is $(v_1, v_2) = |v_1||v_2|cos(\phi) = a \sin x + b \cos x$ where $\phi$ is the angle between $v_1$ and $v_2$, but $|v_1| = \sqrt{a^2 + b^2}, |v_2| = 1$.
