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

Problem

1−cos(x)=1/2

Solution

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

−cos(x)=1/2−1

2. Simplify the right side of the equation.

−cos(x)=−1/2

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

cos(x)=1/2

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

x=π/3

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

x=π/3

x=2*π−π/3=(5*π)/3

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

x=π/3+2*n*π

x=(5*π)/3+2*n*π

Final Answer

x=π/3+2*n*π,(5*π)/3+2*n*π

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