Solve for x 2sin(x)^2-sin(x)-1=0

Problem

2*sin2(x)−sin(x)−1=0

Solution

1. Identify the equation as a quadratic in terms of sin(x)

2. Substitute u=sin(x) to rewrite the equation as a standard quadratic.

2*u2−u−1=0

3. Factor the quadratic expression by finding two numbers that multiply to −2 and add to −1

(2*u+1)*(u−1)=0

4. Solve for u by setting each factor equal 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 using the unit circle.
   For sin(x)=1

x=π/2+2*n*π

For sin(x)=−1/2

x=(7*π)/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

x=π/2+2*n*π,(7*π)/6+2*n*π,(11*π)/6+2*n*π

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