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

$$|a \sin x + b \cos x|=\big|\sqrt{a^2+b^2}\left(\frac a{\sqrt{a^2+b^2}}\sin x+\frac b{\sqrt{a^2+b^2}}\cos x\right)\big|=$$
$$=\sqrt{a^2+b^2}|\left(\sin(x+\phi)\right)|\le\sqrt{a^2+b^2}$$
