SOLUTION: Sin theta . Cos theta = 1/2. Solve the following equation for 0< or equal to x < or equal to 2π
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-> SOLUTION: Sin theta . Cos theta = 1/2. Solve the following equation for 0< or equal to x < or equal to 2π
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Question 1121665: Sin theta . Cos theta = 1/2. Solve the following equation for 0< or equal to x < or equal to 2π Found 2 solutions by htmentor, ikleyn:Answer by htmentor(1343) (Show Source):
You can put this solution on YOUR website! For simplicity, let theta = x
sin(x)cos(x) = 1/2
Using the identity cos(x) = sqrt(1 - sin^2(x)) we have:
sin(x)*sqrt(1 - sin^2(x)) = 1/2
Square both sides:
sin^2(x)(1 - sin^2(x)) = 1/4
sin^4(x) - sin^2(x) + 1/4 = 0
Factor:
(sin^2(x) - 1/2)(sin^2(x) - 1/2) = 0
This gives sin(x) = 1/sqrt(2)
In the interval 0 to 2pi, there are two solutions: x = pi/4 and 5*pi/4
sin(x)*cos(x) = ====> multiply by 2 both sides ====>
2sin(x)*cos(x) = 1 ====> apply the formula 2sin(x)*cos(x) = sin(2x) ====>
sin(2x) = 1 ====>
The solutions are 2x = , 2x = .
First solution gives x = .
Second solution gives x = .
Answer. The original equation has two solutions x = and x = .
