Q. For all values of $\theta$, the values of $3 - \cos{\theta + \cos\left( \theta + \frac{\pi}{3} \right)}$ lie in the interval
A) $\lbrack - 2,\ 3\rbrack$
B) $\lbrack - 2,\ 1\rbrack$
C) [2, 4]
D) [1, 5]
Explanation
(c)
$3 - \cos{\theta + \cos\left( \theta + \frac{\pi}{3} \right)}$
$= 3 - \cos{\theta + \frac{1}{2}\cos{\theta - \frac{\sqrt{3}}{2}}\sin\theta}$
$= 3 - \frac{1}{2}\cos{\theta - \frac{\sqrt{3}}{2}\sin{\theta = 3 - \sin\left( \theta + \frac{\pi}{6} \right)}}$
Since, $- 1 \leq \sin\theta \leq 1$
Hence, the value of expression lies in [2, 4]
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.