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wonderful
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 Trigonometric equation   « on: Jun 15th, 2008, 6:56pm » Quote Modify

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 » Quote Modify

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 » Quote Modify

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 » Quote Modify

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 » Quote Modify

Aryabhatta is correct.

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Sir Col
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 Re: Trigonometric equation   « Reply #5 on: Jun 16th, 2008, 11:38pm » Quote Modify

Stunning, Aryabhatta!
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Aryabhatta
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 Re: Trigonometric equation   « Reply #6 on: Jun 18th, 2008, 7:20pm » Quote Modify

Thanks

I guess we can also do something using complex numbers...
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