Question

The range of values of the expression $5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+1$ is

• A. $[-7,7]$
• B. $[-6,8]$
• C. $[-8,6]$
• D. $[-3\sqrt{3},13]$

Answer 1

Answer: (B)

Let $y=5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+1$
$=5\cos \theta +3\left[\cos \theta \cdot \frac{1}{2}-\sin \theta \frac{\sqrt{3}}{2}\right]+1$
$=\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +1$
Now$\left|\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \right|\le \sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}=7$
$\therefore -7\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le 7$
$\Rightarrow -6\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +1\le 8$
$\Rightarrow y\in [-6,8]$