Solve for x 2cos(x)- square root of 2=0

Problem

2*cos(x)−√(,2)=0

Solution

1. Isolate the trigonometric term by adding √(,2) to both sides of the equation.

2*cos(x)=√(,2)

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

cos(x)=√(,2)/2

3. Identify the reference angle by finding the value of x in the first quadrant where the cosine is √(,2)/2

x=π/4

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

x=π/4

x=2*π−π/4=(7*π)/4

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/4+2*n*π

x=(7*π)/4+2*n*π

Final Answer

x=π/4+2*n*π,(7*π)/4+2*n*π

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