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

Problem

sin(2*x)=cos(x)

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x)

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

2. Subtract cos(x) from both sides to set the equation to zero.

2*sin(x)*cos(x)−cos(x)=0

3. Factor out the common term cos(x)

cos(x)*(2*sin(x)−1)=0

4. Set each factor to zero using the zero product property.

cos(x)=0

2*sin(x)−1=0

5. Solve for x in the first equation cos(x)=0

x=π/2+n*π

6. Solve for x in the second equation 2*sin(x)=1 which simplifies to sin(x)=1/2

x=π/6+2*n*π

x=(5*π)/6+2*n*π

Final Answer

sin(2*x)=cos(x)⇒x=π/2+n*π,π/6+2*n*π,(5*π)/6+2*n*π

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