In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅).sin² x - 2 cos x - 2 = 0

Verified step by step guidance

1. Step 1: Use the Pythagorean identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite the equation in terms of \( \cos x \). Substitute \( \sin^2 x \) with \( 1 - \cos^2 x \) in the equation.
2. Step 2: Simplify the equation to form a quadratic equation in terms of \( \cos x \). This will give you \( 1 - \cos^2 x - 2\cos x - 2 = 0 \).
3. Step 3: Rearrange the terms to form a standard quadratic equation: \( -\cos^2 x - 2\cos x - 1 = 0 \).
4. Step 4: Multiply the entire equation by -1 to make it easier to solve: \( \cos^2 x + 2\cos x + 1 = 0 \).
5. Step 5: Factor the quadratic equation. Notice that it can be factored as \( (\cos x + 1)^2 = 0 \). Solve for \( \cos x \) to find the solutions within the interval \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying trigonometric expressions and solving equations, as they allow for the substitution of one function for another.

Pythagorean Identity

The Pythagorean identity states that for any angle x, sin² x + cos² x = 1. This fundamental relationship between sine and cosine can be rearranged to express one function in terms of the other, which is particularly useful in solving trigonometric equations. In the context of the given equation, it can help transform sin² x into a function of cos x, facilitating the solution process.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation within a specified interval. This often requires the use of identities to rewrite the equation in a more manageable form. Once simplified, solutions can be found by applying inverse trigonometric functions or analyzing the unit circle, ensuring that all solutions fall within the given interval, such as [0, 2π).