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        riddles >> medium >> Find real solutions of the system 
        
(Message started by: tony123 on Oct 23^rd, 2007, 2:52am)

Title: Find real solutions of the system
Post by tony123 on Oct 23^rd, 2007, 2:52am

Find real solutions of the system

sin x + 2 sin(x+y+z) = 0,
sin y + 3 sin(x+y+z) = 0,
sin z + 4 sin(x+y+z) = 0.

Title: Re: Find real solutions of the system
Post by towr on Oct 23^rd, 2007, 3:40am

Let's get the easy ones out of the way: x,y,z multiples of pi works..
More solutions would probably entail doing work, so I'll leave that to someone else for now :P

Title: Re: Find real solutions of the system
Post by pex on Oct 23^rd, 2007, 10:16am

on 10/23/07 at 03:40:35, towr wrote:

Let's get the easy ones out of the way: x,y,z multiples of pi works..
More solutions would probably entail doing work, so I'll leave that to someone else for now :P

~~Having done the work: in fact, there are no other solutions.~~

[hideb]To see this, let
S = [sin x + 2 sin(x+y+z)]² + [sin y + 3 sin(x+y+z)]² + [sin z + 4 sin(x+y+z)]².
We know that S attains the minimum zero: it cannot get any smaller, because it is a sum of squares; and zero is attained by x=y=z=0, for example.

In a minimum, the partial derivatives are zero:
dS/dx = 2[sin x + 2 sin(x+y+z)][cos x + 2 cos(x+y+z)] + 50 sin(x+y+z) cos(x+y+z) = 0
dS/dy = 2[sin y + 3 sin(x+y+z)][cos y + 3 cos(x+y+z)] + 40 sin(x+y+z) cos(x+y+z) = 0
dS/dz = 2[sin z + 4 sin(x+y+z)][cos z + 4 cos(x+y+z)] + 26 sin(x+y+z) cos(x+y+z) = 0
But we know that the minimum is zero; as a consequence, all bold factors are zero at each optimum. This clearly leads to the optimum condition
sin(x+y+z) cos(x+y+z) = 0.

This last condition is satisfied as long as x+y+z is an integer multiple of pi/2. If it is an odd multiple, the first equation of the system becomes
(sin x) + 2 = 0,
which obviously has no solution.

So x+y+z is an even integer multiple of pi/2; that is, an integer multiple of pi. Then, the system of equations simplifies to
sin x = 0
sin y = 0
sin z = 0
and all solutions are precisely given by "x, y and z are integer multiples of pi".[/hideb]

~~Edit: colored text is not hidden...~~

Second Edit: never mind, I should learn how to take derivatives... :-[