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

Problem

sin(x/2)=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 On the unit circle, these angles are π/6 and (5*π)/6

3. Set up the equations for the argument x/2 using the general solution form.

x/2=π/6+2*k*π

x/2=(5*π)/6+2*k*π

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

x=π/3+4*k*π

x=(5*π)/3+4*k*π

Final Answer

sin(x/2)=1/2⇒x=π/3+4*k*π,(5*π)/3+4*k*π

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