In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅).cos 2x + 5 cos x + 3 = 0

Verified step by step guidance

1. Recognize that the equation involves a double angle identity. Recall that \( \cos 2x = 2\cos^2 x - 1 \) or \( \cos 2x = 1 - 2\sin^2 x \). For this problem, use \( \cos 2x = 2\cos^2 x - 1 \).
2. Substitute \( \cos 2x = 2\cos^2 x - 1 \) into the equation, resulting in \( 2\cos^2 x - 1 + 5\cos x + 3 = 0 \).
3. Simplify the equation to form a quadratic in terms of \( \cos x \): \( 2\cos^2 x + 5\cos x + 2 = 0 \).
4. Factor the quadratic equation \( 2\cos^2 x + 5\cos x + 2 = 0 \) into \( (2\cos x + 1)(\cos x + 2) = 0 \).
5. Solve each factor for \( \cos x \): \( 2\cos x + 1 = 0 \) gives \( \cos x = -\frac{1}{2} \) and \( \cos x + 2 = 0 \) gives \( \cos x = -2 \). Since \( \cos x = -2 \) is not possible, find the solutions for \( \cos x = -\frac{1}{2} \) 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, angle sum and difference identities, and double angle identities. In this problem, the double angle identity for cosine, cos(2x) = 2cos²(x) - 1, is particularly useful for transforming the equation into a solvable form.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants. In trigonometric problems, we often convert trigonometric equations into quadratic form to find solutions. After applying the appropriate identities, the equation can be rearranged into a quadratic equation in terms of cos(x), allowing us to use factoring or the quadratic formula to find the values of cos(x).

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this problem, the interval [0, 2π) indicates that we are looking for solutions within one full rotation of the unit circle, including 0 but excluding 2π. Understanding this concept is crucial for determining the valid solutions for x after solving the equation, as we must ensure that all solutions fall within the specified interval.