Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x - sin x - 1 = 0

Verified step by step guidance

1. Recognize that the given equation, \(2 \sin^{2} x - \sin x - 1 = 0\), is quadratic in form with respect to \(\sin x\). To solve it, let \(u = \sin x\) to rewrite the equation as \$2u^{2} - u - 1 = 0$.
2. Solve the quadratic equation \$2u^{2} - u - 1 = 0\( using the quadratic formula: \(u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \)a=2\(, \)b=-1\(, and \)c=-1$.
3. Calculate the discriminant \(\Delta = b^{2} - 4ac\) and then find the two possible values for \(u = \sin x\) from the quadratic formula.
4. For each value of \(\sin x\) found, determine the corresponding values of \(x\) in the interval \([0, 2\pi)\) by using the inverse sine function and considering the unit circle properties (i.e., which quadrants the sine values correspond to).
5. Verify each solution by substituting back into the original equation to ensure they satisfy it, and list all valid solutions for \(x\) within the given interval.

Key Concepts

Quadratic Form in Trigonometric Equations

Some trigonometric equations can be rewritten to resemble quadratic equations by using substitutions like sin²x = (sin x)². Recognizing this form allows the use of algebraic methods such as factoring or the quadratic formula to find solutions for the trigonometric function.

Solving Quadratic Equations

Solving quadratic equations involves finding values of the variable that satisfy the equation, typically by factoring, completing the square, or using the quadratic formula. In trigonometric contexts, these solutions correspond to values of the trigonometric function, which then help find the angle solutions.

Finding Solutions on a Specific Interval [0, 2π)

Trigonometric equations often have infinitely many solutions, so restricting the domain to [0, 2π) means finding all angle solutions within one full rotation. After solving for the trigonometric function, inverse functions and unit circle knowledge are used to determine all valid angles in this interval.