Question

If $f\left(\theta \right)=5\cos \theta +3\cos (\theta +\frac{\pi }{3})+5$, then the range of $f\left(\theta \right)$ is

• A. $[-12,12]$
• B. $[-4,10]$
• C. $[-2,12]$
• D. $[0,12]$

Answer 1

The correct option is C $[-2,12]$

$f\left(\theta \right)=5\cos \theta +3\cos (\theta +\frac{\pi }{3})+5=5\cos \theta +\frac{3}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +5=\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +5$

Now, $-\sqrt{\frac{169}{4}+\frac{27}{4}}\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le \sqrt{\frac{169}{4}+\frac{27}{4}}$
$\Rightarrow -7\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le 7$
$\Rightarrow -7+5\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +5\le 7+5$
$\Rightarrow -2\le f\left(\theta \right)\le 12$

So, the range is $[-2,12]$