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

Problem

cos(2*x)=sin(x)

Solution

1. Apply the double angle identity for cosine to rewrite the left side in terms of sine.

cos(2*x)=1−2*sin2(x)

2. Substitute the identity into the original equation.

1−2*sin2(x)=sin(x)

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

2*sin2(x)+sin(x)−1=0

4. Factor the quadratic expression, treating sin(x) as the variable.

(2*sin(x)−1)*(sin(x)+1)=0

5. 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

6. Solve for x using the unit circle for each case, where n is any integer.

sin(x)=1/2⇒x=π/6+2*n*π

sin(x)=1/2⇒x=(5*π)/6+2*n*π

sin(x)=−1⇒x=(3*π)/2+2*n*π

Final Answer

cos(2*x)=sin(x)⇒x∈{π/6+2*n*π,(5*π)/6+2*n*π,(3*π)/2+2*n*π}

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