Question

If $a\le 3\cos x+5\sin (x-\pi /6)\le b$ for all $x$, then $(a,b)$ is

• A. $(-\sqrt{19},\sqrt{19})$
• B. $(-17,17)$
• C. $(-\sqrt{21},\sqrt{21})$
• D. none of these

Answer 1

The correct option is A $(-\sqrt{19},\sqrt{19})$

$3\cos x+5\sin (x-\frac{\pi }{6})$

$=3\cos x+5\sin x\cos \frac{\pi }{6}-5\cos x\sin \frac{\pi }{6}$

$=3\cos x+\frac{5\sqrt{3}\sin x}{2}-\frac{5\cos x}{2}$

$=\frac{\cos x}{2}+\frac{5\sqrt{3}\sin x}{2}$

Using $-\sqrt{p^2+q^2}\le p\cos x+q\sin x\le \sqrt{p^2+q^2}$

We get

$-\sqrt{19}\le \frac{\cos x}{2}+\frac{5\sqrt{3}\sin x}{2}\le \sqrt{19}$

$\therefore (a,b)=(-\sqrt{19},\sqrt{19})$