Solve for x sin(2x)+sin(4x)=0

Problem

sin(2*x)+sin(4*x)=0

Solution

1. Apply the double angle formula to the term sin(4*x) by treating it as sin(2⋅2*x)

sin(4*x)=2*sin(2*x)*cos(2*x)

2. Substitute this expression back into the original equation.

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

3. Factor out the common term sin(2*x) from the left side.

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

4. Set each factor to zero using the zero product property.

sin(2*x)=0

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

5. Solve the first equation for 2*x using the general solution for sine.

2*x=n*π

x=(n*π)/2

6. Solve the second equation for cos(2*x) and find the general solution for 2*x

cos(2*x)=−1/2

2*x=2*n*π±(2*π)/3

x=n*π±π/3

Final Answer

sin(2*x)+sin(4*x)=0⇒x=(n*π)/2,n*π±π/3

Want more problems? Check here!