SOLUTION: Sin theta . Cos theta = 1/2. Solve the following equation for 0< or equal to x < or equal to 2&#960;

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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
Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
.

There is a simpler way. 


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 = .

Solved.



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