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. insert step 1: Recognize that the equation cos² x - cos x - 1 = 0 is a quadratic equation in terms of cos x.
2. insert step 2: Let y = cos x, then the equation becomes y² - y - 1 = 0.
3. insert step 3: Use the quadratic formula y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to solve for y, where a = 1, b = -1, and c = -1.
4. insert step 4: Calculate the discriminant b² - 4ac and then find the two possible values for y.
5. insert step 5: Use a calculator to find the values of x in the interval [0, 2\pi) that correspond to each solution for y, ensuring the solutions are correct to four decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. In this question, the cosine function is central, as the equation involves cos² x and cos x. Understanding how these functions behave and their values on the unit circle is essential for solving trigonometric equations.

Quadratic Equations

The equation cos² x - cos x - 1 = 0 is a quadratic equation in terms of cos x. Quadratic equations can be solved using various methods, including factoring, completing the square, or the quadratic formula. Recognizing the structure of the equation allows for the application of these methods to find the values of cos x.

Interval Notation

The interval [0, 2π) specifies the range of values for x in which we are interested in finding solutions. This notation indicates that x can take any value from 0 to 2π, including 0 but excluding 2π. Understanding interval notation is crucial for determining the valid solutions of the trigonometric equation within the specified range.