Solve for x sin(2x)=( square root of 2)/2

Problem

sin(2*x)=√(,2)/2

Solution

1. Identify the reference angle for which the sine function equals √(,2)/2

sin(θ)=√(,2)/2⇒θ=π/4

2. Determine the general solutions for the argument 2*x based on the unit circle, where sine is positive in the first and second quadrants.

2*x=π/4+2*k*π

2*x=(3*π)/4+2*k*π

3. Isolate x by dividing both sides of each equation by 2

x=π/8+k*π

x=(3*π)/8+k*π

4. Define the integer constant k to complete the general solution.

k∈ℤ

Final Answer

sin(2*x)=√(,2)/2⇒x=π/8+k*π,(3*π)/8+k*π

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