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

Problem

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

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x) to rewrite the equation.

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

2. Factor out the common term 2*cos(x) from the left side of the equation.

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

3. Set each factor to zero using the zero product property to find the individual equations for x

2*cos(x)=0

1+sin(x)=0

4. Solve the first equation cos(x)=0 for x

x=π/2+n*π

5. Solve the second equation sin(x)=−1 for x

x=(3*π)/2+2*n*π

6. Combine the solutions by noting that the solutions for sin(x)=−1 are already contained within the set of solutions for cos(x)=0

Final Answer

x=π/2+n*π

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