Question

Write the interval in which the value of 5 cos θ + 3 cos $\left(\mathrm{\theta }+\frac{\pi }{3}\right)+3$ lies.

Answer 1

$$\begin{gathered} \mathrm{Let}f\left(\theta \right)=5\cos \theta +3\cos \left(\theta +\frac{\mathrm{\pi }}{3}\right)+3 \\ =5\cos \theta +3(\cos \theta \cos 60°-\sin \theta \sin 60°)+3 \\ =5\cos \theta +\frac{3}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3 \\ =\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3 \\ \mathrm{We}\mathrm{know}\mathrm{that} \\ -\sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}\le \frac{13}{2}\cos \mathrm{\theta }-\frac{3\sqrt{3}}{2}\sin \mathrm{\theta }\le \sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2} \\ -\sqrt{\frac{169}{4}+\frac{27}{4}}\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le \sqrt{\frac{169}{4}+\frac{27}{4}} \\ \Rightarrow -\frac{14}{2}\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le \frac{14}{2} \\ \Rightarrow -7+3\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3\le 7+3 \\ \mathrm{Hence},f\left(\theta \right)\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{interval}\left[-4,10\right]. \end{gathered}$$