Solve for x 4cos(x)+1=2cos(x)

Problem

4*cos(x)+1=2*cos(x)

Solution

1. Subtract 2*cos(x) from both sides of the equation to group the trigonometric terms.

4*cos(x)−2*cos(x)+1=0

2*cos(x)+1=0

2. Isolate the term containing the cosine function by subtracting 1 from both sides.

2*cos(x)=−1

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

cos(x)=−1/2

4. Determine the values of x within the standard interval [0,2*π) where the cosine is −1/2

x=(2*π)/3

x=(4*π)/3

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

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

x=(2*π)/3+2*n*π,(4*π)/3+2*n*π

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