Solve for x sin(x)=2/3

Problem

sin(x)=2/3

Solution

1. Identify the equation as a basic trigonometric equation where the sine of an angle x is equal to a constant value.

2. Apply the inverse sine function to both sides of the equation to find the principal value of x

x=arcsin(2/3)

3. Determine the general solution for x by considering the periodicity of the sine function and the identity sin(π−θ)=sin(θ)

x=arcsin(2/3)+2*π*n

x=π−arcsin(2/3)+2*π*n

4. Define the variable n as any integer to represent all possible solutions.

n∈ℤ

Final Answer

x=arcsin(2/3)+2*π*n,π−arcsin(2/3)+2*π*n

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