SOLUTION: For what value of x does the eq 4sinx+3sin2x-2sin3x+sin4x =2root3

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Question 1040337: For what value of x does the eq 4sinx+3sin2x-2sin3x+sin4x =2root3
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
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For what value of x does the equation 4sinx+3sin2x-2sin3x+sin4x = 2root3
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I can not solve this equation in the strong sense: I can not find all the roots.

But I can guess one root: x = pi%2F6 = 30 deg.


Indeed, then

sin(x)  =  1%2F2;         4sin(x)  =  4%2A%281%2F2%29 = 2;

sin(2x) = sqrt%283%29%2F2;         3sin(2x) = 3%2A%28sqrt%283%29%2F2%29;

sin(3x) = {{1}}};      2sin(3x) = 2;

sin(4x) = sqrt%283%29%2F2%29;          sin(4x) = sqrt%283%29%2F2%29.

Therefore,  4sin(x) + 3sin(2x) - 2sin(3x) + sin(4x) = 2 + 3%2A%28sqrt%283%29%2F2%29 - 2 + sqrt%283%29%2F2 = 4%2A%28sqrt%283%29%2F2%29 = 2sqrt%283%29.    QED.

The problem does not require to find all solution.
It asks to find at least one solution.
One solution is found.
Thus the problem is solved.