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. Start by recalling the Pythagorean identity: \(\sin^{2} x = 1 - \cos^{2} x\). This allows us to rewrite the equation in terms of \(\cos x\) only.
2. Substitute \(\sin^{2} x\) with \(1 - \cos^{2} x\) in the equation: \(1 - \cos^{2} x - 2 \cos x - 2 = 0\).
3. Simplify the equation by combining like terms: \(-\cos^{2} x - 2 \cos x + (1 - 2) = 0\), which simplifies to \(-\cos^{2} x - 2 \cos x - 1 = 0\).
4. Multiply the entire equation by \(-1\) to make the quadratic term positive: \(\cos^{2} x + 2 \cos x + 1 = 0\).
5. Recognize this as a quadratic equation in \(\cos x\). Solve for \(\cos x\) using factoring or the quadratic formula, then find all \(x\) in \([0, 2\pi)\) that satisfy the solutions for \(\cos x\).

Key Concepts

Pythagorean Identity

The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This identity allows you to rewrite expressions involving sin²x in terms of cos²x or vice versa, which is useful for simplifying trigonometric equations.

Trigonometric Equation Solving

Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the trigonometric function. Then, you find all solutions within the given interval by considering the unit circle and periodicity.

Interval Restriction and Solution Sets

When solving trigonometric equations on a specific interval like [0, 2π), it is important to find all angle solutions within that range. This involves understanding the periodic nature of sine and cosine and selecting solutions that fit the interval.