Solve for x 1+sin(x)=2cos(x)^2

Problem

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

Solution

1. Apply the Pythagorean identity to rewrite the equation in terms of a single trigonometric function. Use cos(x)=1−sin(x)

1+sin(x)=2*(1−sin(x))

2. Distribute the constant on the right side of the equation.

1+sin(x)=2−2*sin(x)

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

2*sin(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 by finding the angles that satisfy these sine values within the general solution.

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

x=π/6+2*n*π,(5*π)/6+2*n*π,(3*π)/2+2*n*π

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