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

Problem

2*cos2(x)+cos(x)=0

Solution

1. Factor the common term cos(x) from the left side of the equation.

cos(x)*(2*cos(x)+1)=0

2. Apply the zero product property by setting each factor equal to zero.

cos(x)=0

2*cos(x)+1=0

3. Solve the first equation for x by identifying where the cosine function equals zero on the unit circle.

x=π/2+n*π

4. Isolate the cosine term in the second equation.

2*cos(x)=−1

cos(x)=−1/2

5. Solve the second equation for x by identifying where the cosine function equals −1/2 on the unit circle.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

6. Combine all solutions into a general form where n is an integer.

x=π/2+n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

Final Answer

x=π/2+n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

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