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

Problem

√(,2*cos2(x)+cos(x))=0

Solution

1. Square both sides to eliminate the radical, which results in a quadratic equation in terms of cos(x)

√(,2*cos2(x)+cos(x))2=0

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

2. Factor out the common term cos(x) from the expression.

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

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

cos(x)=0

2*cos(x)+1=0

4. Solve the first case where cos(x)=0 On the unit circle, the cosine is zero at the top and bottom.

x=π/2+n*π

5. Solve the second case by isolating cos(x) in the second factor.

2*cos(x)=−1

cos(x)=−1/2

6. Determine the angles where cos(x)=−1/2 These occur in the second and third quadrants.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

√(,2*cos2(x)+cos(x))=0⇒x=π/2+n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

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