Solve for x sin(4x)=( square root of 3)/2

Problem

sin(4*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 sin(θ)=√(,3)/2 The primary values are θ=π/3 and θ=π−π/3=(2*π)/3

3. Set up the equations for the argument 4*x using the general solution form.

4*x=π/3+2*k*π

4*x=(2*π)/3+2*k*π

4. Isolate x by dividing every term in both equations by 4

x=π/12+(k*π)/2

x=(2*π)/12+(k*π)/2

5. Simplify the fractions in the second set of solutions.

x=π/6+(k*π)/2

Final Answer

x=π/12+(k*π)/2,π/6+(k*π)/2

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