Solve for x sin(x)=1/5

Problem

sin(x)=1/5

Solution

1. Apply the inverse sine function to both sides of the equation to isolate x for the principal value.

arcsin(sin(x))=arcsin(1/5)

2. Determine the principal solution by evaluating the inverse sine.

x=arcsin(1/5)

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

x=π−arcsin(1/5)

4. Account for the periodicity of the sine function by adding integer multiples of 2*π where n is any integer.

x=arcsin(1/5)+2*n*π

x=π−arcsin(1/5)+2*n*π

Final Answer

sin(x)=1/5⇒x=arcsin(1/5)+2*n*π,π−arcsin(1/5)+2*n*π

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