In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 cos 2x + 1 = 0

Verified step by step guidance

1. Start by rewriting the given equation: \(2 \cos 2x + 1 = 0\).
2. Isolate the cosine term by subtracting 1 from both sides and then dividing by 2: \(\cos 2x = -\frac{1}{2}\).
3. Recall that \(\cos \theta = -\frac{1}{2}\) at specific standard angles. Identify all angles \(\theta\) in the interval \([0, 2\pi)\) where this is true. Since the argument is \$2x$, set \(2x = \theta\).
4. Find the general solutions for \$2x\( based on the cosine values, remembering that cosine is negative in the second and third quadrants. Use the reference angle \(\frac{\pi}{3}\) to write the solutions for \)2x$.
5. Finally, solve for \(x\) by dividing all solutions for \$2x\( by 2, and ensure the solutions for \)x$ lie within the interval \([0, 2\pi)\).

Key Concepts

Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of cos(x) or sin(x). It is commonly written as cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity helps simplify or rewrite trigonometric equations involving cos(2x) to solve for x.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given interval. This includes using inverse trigonometric functions and considering the periodicity of sine and cosine to find all valid solutions.

Interval Restriction and Exact vs Approximate Solutions

When solving on a specific interval like [0, 2π), only solutions within that range are valid. Exact values use known special angles (e.g., π/3), while approximate solutions use decimal values rounded to a specified precision, such as four decimal places, to express answers when exact values are not straightforward.