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. Move all terms to one side by adding sin(x) to both sides to set the equation equal 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 at integer multiples of π

x=n*π

6. Solve the second equation for cos(x) by subtracting 1 and dividing by 2.

cos(x)=−1/2

7. Find the angles where the cosine is −1/2 These occur in the second and third quadrants.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

sin(2*x)=−sin(x)⇒x=n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

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