Solve for x sin(2x)=1/2

Problem

sin(2*x)=1/2

Solution

1. Identify the general solution for the sine function. For an equation sin(θ)=a the solutions are θ=arcsin(a)+2*k*π and θ=π−arcsin(a)+2*k*π where k is any integer.

2. Determine the reference angles for which sin(θ)=1/2 These are π/6 and (5*π)/6

3. Set up the equations for the argument 2*x using the reference angles and the period of the sine function.

2*x=π/6+2*k*π

2*x=(5*π)/6+2*k*π

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

x=π/12+k*π

x=(5*π)/12+k*π

Final Answer

sin(2*x)=1/2⇒x=π/12+k*π,(5*π)/12+k*π

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