Solve for x square root of 2cos(x)+1=0

Problem

√(,2*cos(x)+1)=0

Solution

1. Square both sides of the equation to eliminate the square root.

√(,2*cos(x)+1)2=0

2*cos(x)+1=0

2. Isolate the cosine term by subtracting 1 from both sides and then dividing by 2.

2*cos(x)=−1

cos(x)=−1/2

3. Identify the reference angle for which the cosine is positive 1/2

(x_ref)=π/3

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

x=π−π/3=(2*π)/3

x=π+π/3=(4*π)/3

5. Write the general solution by adding 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

√(,2*cos(x)+1)=0⇒x=(2*π)/3+2*n*π,(4*π)/3+2*n*π

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