Question

If $p=4\cos (x-\frac{\pi }{3})+3\sqrt{3}\sin x,$ then the maximum value of $\left[p\right]$ is
(where [.] denotes greatest integer function)

Answer 1

$\cos (x-\frac{\pi }{3})=\cos x\cos \frac{\pi }{3}+\sin x\sin \frac{\pi }{3}$
$\Rightarrow \cos (x-\frac{\pi }{3})=\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x$
So,
$p=4(\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x)+3\sqrt{3}\sin x$
$\Rightarrow p=2\cos x+5\sqrt{3}\sin x$
$\Rightarrow p_{max}=\sqrt{2^2+(5\sqrt{3}{)}^2}=\sqrt{79}$
$\Rightarrow [p{]}_{max}=8$