Question

Determine range $y=3\sin x+4\cos (x+\pi /3)+7$

• A. $7-\sqrt{5-3\sqrt{3}},7+\sqrt{5-3\sqrt{3}}$
• B. $7-\sqrt{10-3\sqrt{3}},7+\sqrt{10-3\sqrt{3}}$
• C. $7-\sqrt{10+3\sqrt{3}},7+\sqrt{10-3\sqrt{3}}$
• D. $7-\sqrt{10+3\sqrt{3}},7-\sqrt{10-3\sqrt{3}}$

Answer 1

The correct option is B $7-\sqrt{10-3\sqrt{3}},7+\sqrt{10-3\sqrt{3}}$

$\cos (x+\frac{\pi }{3})=\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x$
$\therefore y=(3-\frac{\sqrt{3}}{2})\sin x+\frac{1}{2}\cos x+7$ If $3-\frac{\sqrt{3}}{2}=r\cos \alpha$ and $\frac{1}{2}=r\sin \alpha$ then $y=r\sin (x+\alpha )+7$ Range of $\sin (x+\alpha )$ is-1 to 1.$\therefore$ $y=-r+7,r+7$ where $r^2={(3-\frac{\sqrt{3}}{2})}^2+\frac{1}{4}$ or $r^2=9+\frac{3}{4}+\frac{1}{4}-3\sqrt{3}=10-3\sqrt{3}$ or $r=\sqrt{10-3\sqrt{3}}$
$\therefore$ Range is $7-\sqrt{10-3\sqrt{3}},7+\sqrt{10-3\sqrt{3}}$