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   Author  Topic: Trigonometric equation  (Read 961 times)
wonderful
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Trigonometric equation  
« on: Jun 15th, 2008, 6:56pm »
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Solve: (sinx)^2 + (siny)^2 +[sin(x+y)]^2 = 9/4
 
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Barukh
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Re: Trigonometric equation  
« Reply #1 on: Jun 16th, 2008, 11:07am »
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I think calculus may help here.
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Eigenray
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Re: Trigonometric equation  
« Reply #2 on: Jun 16th, 2008, 11:27am »
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I think triangles should help too, but I suck at geometry.
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Aryabhatta
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Re: Trigonometric equation  
« Reply #3 on: Jun 16th, 2008, 3:13pm »
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Pure algebraic manipulations will do too:
 

 
Note that using 1 - 2sin2(x) = cos(2x), we can rewrite the equation as
 
cos (2x) + cos(2y) + cos(2x+2y) = -3/2
 
i.e 2 cos(x+y)cos(x-y) + 2cos2(x+y) = -1/2
 
ie  
1 + 4 cos(x+y)cos(x-y) + 4 cos2(x+y) = 0
 
i.e
 
sin2(x-y) + (cos(x-y) + 2cos(x+y))2 = 0
 
Thus sin(x-y) = 0 and  
cos(x-y) +2 cos(x+y) = 0.
 
The solution set should follow easily now.

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wonderful
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Re: Trigonometric equation  
« Reply #4 on: Jun 16th, 2008, 4:43pm »
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Aryabhatta is correct.
 
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Sir Col
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Re: Trigonometric equation  
« Reply #5 on: Jun 16th, 2008, 11:38pm »
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Stunning, Aryabhatta!  Cool
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Aryabhatta
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Re: Trigonometric equation  
« Reply #6 on: Jun 18th, 2008, 7:20pm »
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Thanks Smiley
 
I guess we can also do something using complex numbers...
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