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

Problem

sin(2*x)−sin(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*sin(x)*cos(x)−sin(x)=0

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

sin(x)*(2*cos(x)−1)=0

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

sin(x)=0

2*cos(x)−1=0

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

x=n*π

5. Solve the second equation 2*cos(x)−1=0 by isolating the cosine function.

cos(x)=1/2

6. Find the general solutions for cos(x)=1/2 using the unit circle.

x=π/3+2*n*π

x=−π/3+2*n*π

Final Answer

sin(2*x)−sin(x)=0⇒x=n*π,π/3+2*n*π,(5*π)/3+2*n*π

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