shubhamsrg
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..
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vikrantg4
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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)
vikrantg4
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@shubhamsrg
shubhamsrg
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i wont guess that'd be a good soln..thats just assumption..
vikrantg4
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lol yeah .. :P
vikrantg4
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@waterineyes
waterineyes
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Thinking..
Ha ha ha.
shubhamsrg
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@mukushla
waterineyes
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offline.
shubhamsrg
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i know,,just tagged him!! :P
shubhamsrg
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maybe @UnkleRhaukus and @ganeshie8
shubhamsrg
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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 ?
shubhamsrg
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@Callisto
waterineyes
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\[\cos(x-y) + \cos(y-z) + \cos(z - x) = 3\]
vishweshshrimali5
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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
waterineyes
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I think you took it seriously @vishweshshrimali5
vishweshshrimali5
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what ? @waterineyes
waterineyes
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\(2k \pi\) is just coming from that fact that general solution of \(cos(x) = cos(y)\)
is given by
\[x = 2n \pi \pm y\]
vishweshshrimali5
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yes......... trigonometric equations
waterineyes
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Yes..
vishweshshrimali5
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we can write sin x = cos(90-x)........ will that help ?
waterineyes
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Let me remind the formula for : \((a + b + c)^2\)
vishweshshrimali5
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Yes.............. \(\large a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
waterineyes
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Okay..
waterineyes
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Check subhamarsh solution he is right in calculating this..
shubhamsrg
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????
waterineyes
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Nothing..
mukushla
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guys u already got it...
mukushla
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\[\cos(x-y)+\cos(x-z)+\cos(y-z) \le1+1+1=3\]equality occurs when\[cos(x-y)=\cos(x-z)=\cos(y-z)=1\]
shubhamsrg
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@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 ( )
waterineyes
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That will not matter I think:
\[\cos(-\theta) = \cos(\theta)\]
waterineyes
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Oh wow....
waterineyes
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I understand the feelings..
Ha ha ha...
shubhamsrg
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sorry didnt catch ya ..come again..
shubhamsrg
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x-y = 2npi ,, x-z = 2mpi,,
after that..
mukushla
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\[y=x-2n\pi\]\[z=x-2m\pi\] put them in the first one
\[\cos x+\cos (x-2n\pi)+\cos(x-2m\pi)=3\cos 30\]\[\cos x+\cos x+\cos x=3\cos 30\]
shubhamsrg
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aha..i see..thanks again..
mukushla
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welcome