Sorry, I don't know if this is in the right section or not.
Basically, the question is:
Solve sin(x) + sin(y) + sin(z) = 0 , for all x,y and z.
How I attempted it:
I assumed the identity sin(x) + sin(x+2pi/3) + sin(x-2pi/3) = 0 , which is easily proved with the compound angle formulae.
Then I just compared the equation and the identity, and said that if y were of the form x+2pi/3 and z were of the form x-2pi/3, then the equation would hold. So effectively there are an infinite number of solutions.
The next question was:
Solve cos(x) + cos(y) + sin(z) = 0
FOr this, I re-wrote the cos's as sin(Pi/2-x) (or y, as the case may be), and solved it exactly the same way as above.
Are these methods correct? I'm filled wiht doubt because on another forum they were talking about putting them in the complex plane as vectors and rotating them and all sorts of things...
To the guy I quoted: 'A punch in the wind'? Is that good or bad :S Anyway, no other information was given, and I'm not too interested in getting every single solution out - as long as my method -would- get every solution out I'll be happy.
There is an almost impossible target. It is out there but you have no idea where it is. It might be beyond your reach or within your reach. You have to do something. Out of desperation, and with some prayer, you throw a punch in the wind, hoping you'd hit the target.
A billion punches, in all conceivable directions, might not even scratch your target.