Solve for x sin(x)=1/3

Problem

sin(x)=1/3

Solution

1. Identify the general solution for the sine function, which is given by x=arcsin(a)+2*n*π and x=π−arcsin(a)+2*n*π for any integer n

2. Apply the inverse sine to find the principal value.

arcsin(1/3)

3. Write the first set of solutions using the principal value and the period of the sine function.

x=arcsin(1/3)+2*n*π

4. Write the second set of solutions by using the symmetry property of sine, sin(π−θ)=sin(θ)

x=π−arcsin(1/3)+2*n*π

Final Answer

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

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