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. 7 cos x = 4 - 2 sin² x

Verified step by step guidance

1. Rewrite the given equation: \(7 \cos x = 4 - 2 \sin^{2} x\).
2. Use the Pythagorean identity \(\sin^{2} x = 1 - \cos^{2} x\) to express everything in terms of \(\cos x\). Substitute to get: \(7 \cos x = 4 - 2(1 - \cos^{2} x)\).
3. Simplify the right side: \(7 \cos x = 4 - 2 + 2 \cos^{2} x\), which becomes \(7 \cos x = 2 + 2 \cos^{2} x\).
4. Rearrange the equation to standard quadratic form in terms of \(\cos x\): \(2 \cos^{2} x - 7 \cos x + 2 = 0\).
5. Solve the quadratic equation for \(\cos x\) using the quadratic formula \(\cos x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a=2\), \(b=-7\), and \(c=2\). Then find all \(x\) in \([0, 2\pi)\) such that \(\cos x\) equals the solutions found.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, the Pythagorean identity sin²x + cos²x = 1 is essential to rewrite sin²x in terms of cos²x, enabling the equation to be expressed in a single trigonometric function for easier solving.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This often requires algebraic manipulation, use of identities, and understanding the periodic nature of sine and cosine to find all valid solutions between 0 and 2π.

Interval and Exact vs Approximate Solutions

The problem restricts solutions to the interval [0, 2π), meaning only angles within one full rotation are considered. Solutions should be given as exact values (like π/3) when possible, or approximated to four decimal places when exact forms are complicated or unavailable, ensuring clarity and precision.