Solve for x sin(2x)=( square root of 3)/2

Problem

sin(2*x)=√(,3)/2

Solution

1. Identify the general solutions 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 the sine is √(,3)/2 These values are π/3 and (2*π)/3

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

2*x=π/3+2*k*π

2*x=(2*π)/3+2*k*π

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

x=π/6+k*π

x=π/3+k*π

Final Answer

sin(2*x)=√(,3)/2⇒x=π/6+k*π,π/3+k*π

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