Solve for x 2cos(x)-1=0

Problem

2*cos(x)−1=0

Solution

1. Isolate the trigonometric term by adding 1 to both sides of the equation.

2*cos(x)=1

2. Divide both sides by 2 to solve for cos(x)

cos(x)=1/2

3. Identify the reference angle by finding the value of x in the first quadrant where the cosine is 1/2

x=π/3

4. Determine all solutions within the interval [0,2*π) Since cosine is positive in the first and fourth quadrants, the solutions are x=π/3 and x=2*π−π/3

x=π/3,(5*π)/3

5. Generalize the solution by adding multiples of the period 2*π to account for all possible rotations, 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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