Question

Find the range of $f\left(x\right)=3\sin x+2\cos x+3$

Answer 1

$3\sin x+2\cos x+3$

$=\sqrt{3^2+2^2}\sin (x+\varphi )+3$

$f\left(x\right)=\sqrt{13}\sin (x+\varphi )+3$

$-1\le \sin (x+\varphi )\le 1$

$\Rightarrow -\sqrt{13}\le \sqrt{13}\sin (x+\varphi )\le \sqrt{13}$

$\Rightarrow 3-\sqrt{13}\le \sqrt{13}\sin (x+\varphi )+3\le 3+\sqrt{13}$

Hence Range$\in [3-\sqrt{13},3+\sqrt{13}]$