Solve for x sin(x)=1/3

Problem

sin(x)=1/3

Solution

1. Identify the equation as a basic trigonometric equation of the form sin(x)=a

2. Find the principal value by taking the inverse sine of both sides to find the first solution in the interval [−π/2,π/2]

x=arcsin(1/3)

3. Determine the second solution within one period [0,2*π) by using the symmetry property of the sine function, sin(π−x)=sin(x)

x=π−arcsin(1/3)

4. Generalize the solutions to account for the periodicity of the sine function by adding 2*π*n where n is any integer.

x=arcsin(1/3)+2*π*n

x=π−arcsin(1/3)+2*π*n

Final Answer

sin(x)=1/3⇒x=arcsin(1/3)+2*π*n,π−arcsin(1/3)+2*π*n

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