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)

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.

sin(x)=0

2*cos(x)+1=0

4. Solve the first equation for x The sine function is zero at integer multiples of π

x=n*π

5. Solve the second equation for cos(x)

cos(x)=−1/2

6. Find the values of x 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

x=n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

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