In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos² x - cos x - 1 = 0

Verified step by step guidance

1. Recognize that the equation is a quadratic in terms of \( \cos x \). Let \( y = \cos x \), so the equation becomes \( y^2 - y - 1 = 0 \).
2. Use the quadratic formula to solve for \( y \): \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -1 \).
3. Calculate the discriminant \( \Delta = b^2 - 4ac = (-1)^2 - 4(1)(-1) = 1 + 4 = 5 \), then find the two roots \( y_1 \) and \( y_2 \).
4. Since \( y = \cos x \), check which roots are within the valid range for cosine values, i.e., between -1 and 1. Discard any root outside this range.
5. For each valid root \( y \), solve \( \cos x = y \) on the interval \( [0, 2\pi) \) using the inverse cosine function and symmetry properties of cosine, then use a calculator to find the approximate values of \( x \) correct to four decimal places.

Key Concepts

Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. Solving these equations means finding all angle values within a specified interval that satisfy the equation. Understanding how to manipulate and rearrange these equations is essential for isolating the trigonometric function.

Quadratic Form in Trigonometry

Some trigonometric equations can be rewritten as quadratic equations by substituting a trigonometric function (e.g., cos x) with a variable. This allows the use of algebraic methods like factoring or the quadratic formula to find solutions for the trigonometric function before solving for the angle.

Using a Calculator and Interval Restrictions

After finding the trigonometric function values, a calculator is used to determine the corresponding angles, ensuring answers are accurate to four decimal places. Solutions must be restricted to the given interval [0, 2π), meaning only angles within one full rotation are considered.