How do we find maximum value of $a$?

Question

For $n$ real numbers $x_1,x_2,\ldots,x_n$ ( $n\geq 2$), consider

$\displaystyle a=\sin(x_1)\cos(x_2)+\sin(x_2)\cos(x_3)+\cdots+\sin(x_n)\cos(x_1).$

• How do we find maximum value of $a$?

• For what values of $x_1, x_2...., x_n$ is the maximum achieved?

I want to get help from you If that's possible.

Regards

Answer 1

By C-S and AM-GM we obtain: $$\sum_{k=1}^n\sin{x_k}\cos{x_{k+1}}\leq\sqrt{\sum_{k=1}^n\sin^2x_k\sum_{k=1}^n\cos^2x_k}=$$ $$=\sqrt{\sum_{k=1}^n\sin^2x_k\left(n-\sum_{k=1}^n\sin^2x_k\right)}\leq\frac{n}{2}.$$ The equality occurs for $x_k=\frac{\pi}{4}$, which says that $\frac{n}{2}$ is a maximal value.