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

Use Cauchy-Schwarz inequality :
\begin{align}|a\sin x+b \cos x| = |(a,b)\cdot (\sin x,\cos x)|
&\leq \sqrt{a^2+b^2}\sqrt{\sin^2 x+\cos^2 x}=\sqrt{a^2+b^2}.\end{align}
