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

Problem

2*cos(x)+1=0

Solution

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

2*cos(x)=−1

2. Divide both sides by 2 to solve for the cosine of x

cos(x)=−1/2

3. Identify the reference angle in the first quadrant where cos(θ)=1/2 which is π/3 (or 60.

4. Determine the quadrants where the cosine function is negative, which are Quadrant II and Quadrant III.

5. Calculate the specific solutions within the interval [0,2*π) using the reference angle.

x=π−π/3=(2*π)/3

x=π+π/3=(4*π)/3

6. Generalize the solution by adding multiples of the period 2*π to account for all possible values of x

x=(2*π)/3+2*k*π

x=(4*π)/3+2*k*π

Final Answer

x=(2*π)/3+2*k*π,(4*π)/3+2*k*π

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