Maximize $x\cos \left( {{\theta _a}} \right) + y\cos \left( {{\theta _b}} \right)$ subject to $x\le 3$, $y\le5$?

Question

I like to maximize the following equation:

$x\cos \left( {{\theta _a}} \right) + y\cos \left( {{\theta _b}} \right)$ subject to $x\le 3$, $y\le5$

Where, $cos(\theta_a)$ and $cos(\theta_b)$ are just two angels range $0\le \theta_a \le 2\pi $ and $0\le \theta_b \le 2\pi $

By doing numerically I found that $x=3$ and $y=5$ maximize the equation. But how should I determine the condition on the angle $\theta_a$ and $\theta_b$?

Any suggestions will be helpful. Thanks.

Answer 1

Since $x$ and $y$ are positive numbers, you want $\cos \theta_a$ and $\cos \theta_b$ to be as large as possible, so take $\theta_a=\theta_b=0$