If $A+B+C+D+E = 540^\\circ$ what is $\\min (\\cos A+\\cos B+\\cos C+\\cos D+\\cos E)$?

Question

Let each of $A, B, C, D, E$ be an angle that is less than $180^\\circ$ and is greater than $0^\\circ$. Note that each angle can be neither $0^\\circ$ nor $180^\\circ$. If $A+B+C+D+E = 540^\\circ,$ what is the minimum of the following function? $$\\cos A+\\cos B+\\cos C+\\cos D+\\cos E$$ I suspect the minimum is achieved when $A=B=C=D=E$, but I can't prove it. I need your help.

Answer 1

Yes, you are correct in your suspicion. (I ignored all the \\ and $ symbols in your message.)

First, notice that if all 5 angles must total 540, then the average of the 5 angles must be 540/5 = 108.

One way for the average of the angles to equal 108 is for them all to simply be = 108.

Next envision the unit circle. Where is 108 on the unit circle?

108 is located in Quadrant II.

Now, they want to know what angles will minimize this function given, which is the sum of the cosine of each angle.

How would we minimize that function? By minimizing cosine of each angle.

Looking at 108 on the unit circle, we can see that moving towards 180 would decrease the cosine of the angle, moving the function closer to the minimum.

So, that sum of cosines function is going to be minimized when all angles are as close as possible to 180.

First, evaluate the function using all angles of 108, so we have something to compare to.

This is = 5*cos(108) = -1.545084972

Now, we can spread out the angles on the unit circle, but they still must average 108.

So the question is, can we spread out the angles in such a way that they still average 108, but are closer to 180, to decrease that sum of cosines function further?

Test it out. Keep 3 angles at 108, and let's move the last two angles slightly so that they still average 108, and see what the sum of cosines function evaluates to. So, A=108, B=108, C=108, D=109, E=107.

In the sum of cosines function, this gives = -1.544990842

This number is slightly larger than before. This demonstrates that selecting angles more spread out from 108 decreases that sum of cosines function.

Therefore, the minimum of the function must be when A=B=C=D=E=108.