The value of $5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3$ lies between

The value of $5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3$ lies between
  1. -2 and 5
  2. -1 and 8
  3. -3 and 6
  4. -4 and 10

Solution

Let $f(\theta)=5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{2}\right)+3$ $\Rightarrow f(\theta)=5 \cos \theta+3 \cos \theta \cdot \frac{1}{2}-3 \sin \theta \cdot \frac{\sqrt{3}}{2}+3$ $=\frac{13}{2} \cos \theta-\frac{3 \sqrt{3}}{2} \sin \theta+3$ $\begin{aligned} & \text { Since, }-\sqrt{\frac{169}{4}+\frac{27}{4}} \leq \frac{13}{2} \cos \theta-\frac{3 \sqrt{3}}{2} \sin \theta \leq \sqrt{\frac{169}{4}+\frac{27}{4}} \\ & \Rightarrow-\frac{14}{2}+3 \leq \frac{13}{2} \cos \theta-\frac{3 \sqrt{3}}{2} \sin \theta+3 \leq \frac{14}{2}+3 \\ & \Rightarrow-4 \leq f(\theta) \leq 10\end{aligned}$

Asked in: AP EAMCET 2024 (19 May Shift 2)

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