Solve by Factoring 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 first term.

2*sin(x)*cos(x)−sin(x)=0

2. Factor out the greatest common factor, which is sin(x) from the left side of the equation.

sin(x)*(2*cos(x)−1)=0

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

sin(x)=0

2*cos(x)−1=0

4. Solve for x in the first equation sin(x)=0 This occurs at integer multiples of π

x=n*π

5. Isolate the cosine function in the second equation 2*cos(x)−1=0

cos(x)=1/2

6. Solve for x in the equation cos(x)=1/2 This occurs at π/3 and (5*π)/3 within the first rotation, plus any integer multiple of 2*π

x=π/3+2*n*π

x=(5*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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