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

Problem

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

Solution

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

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

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

cos(x)=0

2*sin(x)+1=0

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

x=π/2,(3*π)/2

4. Isolate the sine function in the second equation.

2*sin(x)=−1

sin(x)=−1/2

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

x=(7*π)/6,(11*π)/6

6. Generalize the solution by adding 2*π*n to account for the periodicity of the functions, where n is an integer.

x=π/2+π*n

x=(7*π)/6+2*π*n

x=(11*π)/6+2*π*n

Final Answer

x=π/2+π*n,(7*π)/6+2*π*n,(11*π)/6+2*π*n

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