Question

Write the interval in which the values of $5cos\theta +3cos(\theta +\frac{\pi }{3})+3$ lie.

Answer 1

Let $f\left(\theta \right)=5cos\theta +3cos(\theta +\frac{\pi }{3})+3$then
$f\left(\theta \right)=5cos\theta +3cos(\theta +\frac{\pi }{3})+3$
$=5cos\theta +3(cos\theta cos\frac{\pi }{3}-sin\theta sin\frac{\pi }{3})+3$
$=5cos\theta +\frac{3}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +3$
$=\frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +3$ ...(i)
Now,
$-\sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}\le \frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta \le \sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}$
$\Rightarrow -7\le \frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +3\le 7+3$
$\Rightarrow -4\le \frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +3\le 10$
$\Rightarrow -4\le 5cos\theta -3cos(\theta +\frac{\pi }{3})+3\le 10$ [Using 1] Hence, the required interval is [-4,10].