Solve for x cos(x)-2sin(x)cos(x)=0

Problem

cos(x)−2*sin(x)*cos(x)=0

Solution

1. Identify the common factor in the terms on the left side of the equation.

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

cos(x)*(1−2*sin(x))=0

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

cos(x)=0

1−2*sin(x)=0

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

x=π/2,(3*π)/2

5. Isolate the sine function in the second equation.

−2*sin(x)=−1

sin(x)=1/2

6. Solve for x where the sine value is 1/2 within the standard interval [0,2*π)

x=π/6,(5*π)/6

7. Generalize the solution by adding 2*π*n to account for the periodicity of the functions, where n is any integer. Note that π/2 and (3*π)/2 can be combined into a single expression.

Final Answer

x=π/2+π*n,π/6+2*π*n,(5*π)/6+2*π*n

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