Q67 / 160 · TS EAMCET 2006 · Mathematics

For all values of $\theta$, the values of $3-\cos \theta+\cos \left(\theta+\frac{\pi}{3}\right)$ lie in the interval :

  • A
    $[-2,3]$
  • B
    $[-2,1]$
  • Correct answer
    C
    $[2,4]$
  • D
    $[1,5]$

Solution

$3-\cos \theta+\cos \left(\theta+\frac{\pi}{3}\right)$ $=3-\cos \theta+\cos \theta \cdot \cos \frac{\pi}{3}-\sin \theta \cdot \sin \frac{\pi}{3}$ $=3-\cos \theta+\frac{1}{2} \cos \theta-\frac{\sqrt{3}}{2} \sin \theta$ $=3-\frac{1}{2} \cos \theta-\frac{\sqrt{3}}{2} \sin \theta$ $=3-\left[\sin \frac{\pi}{6} \cos \theta+\cos \frac{\pi}{6} \sin \theta\right]$ $=3-\sin \left(\theta+\frac{\pi}{6}\right)$ Since $-1 \leq \sin \theta \leq 1$ $\therefore$ The value of expression lies in $[2,4]$.

Hence, the correct answer is option: 'C'.

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