Solve for x 1-sin(x)=1/2

Problem

1−sin(x)=1/2

Solution

1. Isolate the sine term by subtracting 1 from both sides of the equation.

−sin(x)=1/2−1

2. Simplify the right side of the equation.

−sin(x)=−1/2

3. Multiply both sides by −1 to solve for sin(x)

sin(x)=1/2

4. Identify the reference angle in the first quadrant where the sine value is 1/2

x=π/6

5. Determine all solutions within the standard interval [0,2*π) by finding where sine is positive (Quadrants I and II).

x=π/6

x=π−π/6=(5*π)/6

6. Generalize the solution by adding multiples of the period 2*π where n is any integer.

x=π/6+2*n*π

x=(5*π)/6+2*n*π

Final Answer

x=π/6+2*n*π,(5*π)/6+2*n*π

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