Solve for x y=sin(3x-2pi)

Problem

y=sin(3*x−2*π)

Solution

1. Apply periodicity of the sine function. Since sin(θ−2*π)=sin(θ) the equation simplifies immediately.

y=sin(3*x)

2. Take the inverse sine of both sides to begin isolating the variable x

arcsin(y)=3*x

3. Isolate x by dividing both sides of the equation by 3

x=arcsin(y)/3

4. Account for the general solution of the sine function. Because sine is periodic, the general solution for 3*x is n*π+(−1)n*arcsin(y)

3*x=n*π+(−1)n*arcsin(y)

5. Divide by 3 to find the general expression for x where n is any integer.

x=(n*π+(−1)n*arcsin(y))/3

Final Answer

x=(n*π+(−1)n*arcsin(y))/3

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