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  4. The range of values of the expression 5 cos θ+3 cos (θ+(π/3))+1 is

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

Trigonometric Functions

Solution:

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