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

Problem

sin(x)=sin(2*x)

Solution

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

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

2. Rearrange the equation by subtracting 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. Set each factor to zero using the zero product property to find the possible solutions for x

sin(x)=0

2*cos(x)−1=0

5. Solve the first equation sin(x)=0 This occurs when x is an integer multiple of π

x=n*π

6. Solve the second equation by isolating cos(x)

2*cos(x)=1

cos(x)=1/2

7. Determine the values for x where cos(x)=1/2 This occurs at π/3 and (5*π)/3 within the first rotation, or more generally:

x=π/3+2*n*π

x=(5*π)/3+2*n*π

Final Answer

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

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