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        riddles >> medium >> Trigonometric equation
        
(Message started by: wonderful on Jun 15th, 2008, 6:56pm)
Title: Trigonometric equation
Post by wonderful on Jun 15th, 2008, 6:56pm
Solve: (sinx)^2 + (siny)^2 +[sin(x+y)]^2 = 9/4

Have A Great Day!
Title: Re: Trigonometric equation
Post by Barukh on Jun 16th, 2008, 11:07am
I think [hide]calculus[/hide] may help here.
Title: Re: Trigonometric equation
Post by Eigenray on Jun 16th, 2008, 11:27am
I think [hide]triangles[/hide] should help too, but I suck at [hide]geometry[/hide].
Title: Re: Trigonometric equation
Post by Aryabhatta on Jun 16th, 2008, 3:13pm
Pure algebraic manipulations will do too:

[hide]

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.
[/hide]
Title: Re: Trigonometric equation
Post by wonderful on Jun 16th, 2008, 4:43pm
Aryabhatta is correct.

Have A Great Day!
Title: Re: Trigonometric equation
Post by Sir Col on Jun 16th, 2008, 11:38pm
Stunning, Aryabhatta!  8)
Title: Re: Trigonometric equation
Post by Aryabhatta on Jun 18th, 2008, 7:20pm
Thanks :)

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


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