Solve for x 2sin(2x)cos(x)-sin(x)=0

Problem

2*sin(2*x)*cos(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*(2*sin(x)*cos(x))*cos(x)−sin(x)=0

2. Simplify the expression by multiplying the constants and the cosine terms.

4*sin(x)*cos2(x)−sin(x)=0

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

sin(x)*(4*cos2(x)−1)=0

4. Set each factor to zero using the zero product property to find the possible values for x

sin(x)=0

4*cos2(x)−1=0

5. Solve the first equation sin(x)=0

x=n*π

6. Solve the second equation by isolating cos2(x) and taking the square root.

4*cos2(x)=1

cos2(x)=1/4

cos(x)=±1/2

7. Determine the values of x where cos(x)=1/2 and cos(x)=−1/2

x=π/3+2*n*π

x=(5*π)/3+2*n*π

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

8. Combine the solutions for cos(x)=±1/2 into a more compact form.

x=π/3+n*π

x=(2*π)/3+n*π

Final Answer

x=n*π,π/3+n*π,(2*π)/3+n*π

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