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

Problem

cos(2*x)=sin(x)

Solution

1. Apply the double angle identity for cos(2*x) to rewrite the equation in terms of sin(x) using the formula cos(2*x)=1−2*sin2(x)

1−2*sin2(x)=sin(x)

2. Rearrange the equation into a standard quadratic form by moving all terms to one side.

2*sin2(x)+sin(x)−1=0

3. Factor the quadratic expression by treating sin(x) as the variable.

(2*sin(x)−1)*(sin(x)+1)=0

4. Set each factor to zero to find the possible values for sin(x)

2*sin(x)−1=0⇒sin(x)=1/2

sin(x)+1=0⇒sin(x)=−1

5. Solve for x using the unit circle values and general solutions for sine.
   For sin(x)=1/2

x=π/6+2*k*π

x=(5*π)/6+2*k*π

For sin(x)=−1

x=(3*π)/2+2*k*π

Final Answer

cos(2*x)=sin(x)⇒x=π/6+2*k*π,(5*π)/6+2*k*π,(3*π)/2+2*k*π

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