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

Problem

2*sin2(x)+sin(x)=1

Solution

1. Rearrange the equation into a standard quadratic form by subtracting 1 from both sides.

2*sin2(x)+sin(x)−1=0

2. Substitute a variable to simplify the quadratic, letting u=sin(x)

2*u2+u−1=0

3. Factor the quadratic expression.

(2*u−1)*(u+1)=0

4. Solve for u by setting each factor to zero.

2*u−1=0⇒u=1/2

u+1=0⇒u=−1

5. Back-substitute sin(x) for u to find the values of x

sin(x)=1/2

sin(x)=−1

6. Determine the general solutions for x based on the unit circle.
   For sin(x)=1/2

x=π/6+2*n*π

x=(5*π)/6+2*n*π

For 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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