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Extreme values and Periodicity of Trigonometric functions

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Question

The maximum value of 5 $\cos θ + 3 \cos (θ + \frac{π}{3}) + 3$ is

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Solution

We have,
$\begin{array}{l} 5 \cos θ + 3 \cos (θ + \frac{π}{3}) + 3 \\ = 5 \cos θ + 3 \cos θ \cos \frac{π}{3} - 3 \sin θ \sin \frac{π}{3} + 3 \\ = \frac{13}{2} \cos θ - \frac{3 \sqrt{3}}{2} \sin θ + 3 \end{array}$
Now,
$\begin{array}{l} - \sqrt{{(\frac{13}{2})}^{2}} + {(\frac{3 \sqrt{3}}{2})}^{2} \leq \frac{13}{2} \cos θ - \frac{3 \sqrt{3}}{2} \sin θ \leq \sqrt{{(\frac{13}{2})}^{2} + {(\frac{3 \sqrt{3}}{2})}^{2}} \end{array}$
$\begin{array}{l} \Rightarrow - 7 \leq \frac{13}{2} \cos θ - \frac{3 \sqrt{3}}{2} \sin θ \leq 7 for all θ \\ \Rightarrow - 4 \leq \frac{13}{2} \cos θ - \frac{3 \sqrt{3}}{2} \sin θ + 3 \leq 10 for all θ \\ \Rightarrow - 4 \leq 5 \cos θ + 3 \cos (θ + \frac{π}{3}) + 3 \leq 10 for all θ \end{array}$

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