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

Problem

sin(2*x)=sin(x)

Solution

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

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

2. Subtract sin(x) from both sides to set the equation to zero.

2*sin(x)*cos(x)−sin(x)=0

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

sin(x)*(2*cos(x)−1)=0

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

sin(x)=0

2*cos(x)−1=0

5. Solve the first equation for x by identifying where the sine function is zero.

x=n*π

6. Solve the second equation for x by isolating the cosine term.

cos(x)=1/2

x=π/3+2*n*π

x=−π/3+2*n*π

7. Combine the solutions into a general form where n is any integer.

Final Answer

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

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