shubhamsrg 3 years ago if cos x + cos y + cos z = 3 sin60 and sin x + sin y + sin z = 3 cos60 then how do i prove that x=pi/6 + 2k(pi) y=pi/6 + 2l(pi) and z=pi/6 + 2m(pi) where k,l,m are integers..

1. vikrantg4

i dont have perfect solution.. still i logically say that cos x +cos y + cos z = cos 30 + cos 30 + cos 30 which means x=y=z=pi/6 since sin and cos repeat after intervals of 2pi x = y = z = pi/6 + 2n(pi)

2. vikrantg4

@shubhamsrg

3. shubhamsrg

i wont guess that'd be a good soln..thats just assumption..

4. vikrantg4

lol yeah .. :P

5. vikrantg4

@waterineyes

6. waterineyes

Thinking.. Ha ha ha.

7. shubhamsrg

@mukushla

8. waterineyes

offline.

9. shubhamsrg

i know,,just tagged him!! :P

10. shubhamsrg

maybe @UnkleRhaukus and @ganeshie8

11. shubhamsrg

i sq both and added them,,got cosx cosy + sinx siny + cosx cosz + sinx sinz + cosz cosy + sinz siny = 3 so its cos( ) + cos ( ) + cos ( ) = 3 where space inside ( ) denotes something now each cos ( ) = 1 but how do i reach my ans ?

12. shubhamsrg

@Callisto

13. waterineyes

$\cos(x-y) + \cos(y-z) + \cos(z - x) = 3$

14. vishweshshrimali5

well I recall something from $$2k\pi$$ and that is demoirve's theorem from complex numbers which has a lot to do with such questions

15. waterineyes

I think you took it seriously @vishweshshrimali5

16. vishweshshrimali5

what ? @waterineyes

17. waterineyes

$$2k \pi$$ is just coming from that fact that general solution of $$cos(x) = cos(y)$$ is given by $x = 2n \pi \pm y$

18. vishweshshrimali5

yes......... trigonometric equations

19. waterineyes

Yes..

20. vishweshshrimali5

we can write sin x = cos(90-x)........ will that help ?

21. waterineyes

Let me remind the formula for : $$(a + b + c)^2$$

22. vishweshshrimali5

Yes.............. $$\large a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

23. waterineyes

Okay..

24. vishweshshrimali5

Now ?

25. waterineyes

Check subhamarsh solution he is right in calculating this..

26. shubhamsrg

????

27. waterineyes

Nothing..

28. mukushla

29. mukushla

$\cos(x-y)+\cos(x-z)+\cos(y-z) \le1+1+1=3$equality occurs when$cos(x-y)=\cos(x-z)=\cos(y-z)=1$

30. shubhamsrg

@mukushla how do we reach the ans from here?? also we dont know if it'll be x-y or y-x under ( ),,same argument for other ( )

31. waterineyes

That will not matter I think: $\cos(-\theta) = \cos(\theta)$

32. waterineyes

Oh wow....

33. waterineyes

I understand the feelings.. Ha ha ha...

34. shubhamsrg

sorry didnt catch ya ..come again..

35. shubhamsrg

x-y = 2npi ,, x-z = 2mpi,, after that..

36. mukushla

$y=x-2n\pi$$z=x-2m\pi$ put them in the first one $\cos x+\cos (x-2n\pi)+\cos(x-2m\pi)=3\cos 30$$\cos x+\cos x+\cos x=3\cos 30$

37. shubhamsrg

aha..i see..thanks again..

38. mukushla

welcome