Solve for x 2cos(x)^2- square root of 3cos(x)=0

Problem

2*cos2(x)−√(,3)*cos(x)=0

Solution

1. Identify the common factor in the expression, which is cos(x)

2. Factor out cos(x) from both terms on the left side of the equation.

cos(x)*(2*cos(x)−√(,3))=0

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

cos(x)=0

2*cos(x)−√(,3)=0

4. Solve the first equation for x within the standard interval [0,2*π)

cos(x)=0⇒x=π/2,(3*π)/2

5. Isolate the trigonometric function in the second equation.

2*cos(x)=√(,3)

cos(x)=√(,3)/2

6. Solve the second equation for x within the standard interval [0,2*π)

cos(x)=√(,3)/2⇒x=π/6,(11*π)/6

7. Generalize the solutions by adding 2*π*n to account for the periodicity of the cosine function, where n is any integer.

x=π/2+π*n

x=π/6+2*π*n

x=(11*π)/6+2*π*n

Final Answer

x=π/2+π*n,π/6+2*π*n,(11*π)/6+2*π*n

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