In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + cos x = 1

Verified step by step guidance

1. Start by rewriting the given equation: \(2 \sin^{2} x + \cos x = 1\).
2. Use the Pythagorean identity \(\sin^{2} x = 1 - \cos^{2} x\) to express the equation entirely in terms of \(\cos x\). Substitute to get: \(2(1 - \cos^{2} x) + \cos x = 1\).
3. Simplify the equation: \(2 - 2 \cos^{2} x + \cos x = 1\). Then rearrange to form a quadratic equation in \(\cos x\): \(-2 \cos^{2} x + \cos x + 1 = 0\).
4. Multiply the entire equation by \(-1\) to make the quadratic standard: \(2 \cos^{2} x - \cos x - 1 = 0\). Now solve this quadratic equation for \(\cos x\) using the quadratic formula \(\cos x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a=2\), \(b=-1\), and \(c=-1\).
5. After finding the values of \(\cos x\), determine the corresponding values of \(x\) in the interval \([0, 2\pi)\) by using the inverse cosine function and considering the cosine sign in different quadrants.

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 cos x, simplifying the equation for easier solving.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This requires understanding how to manipulate the equation algebraically and use inverse trigonometric functions to find exact or approximate angle values.

Interval Notation and Solution Sets

The problem restricts solutions to the interval [0, 2π), meaning all solutions must be found within one full rotation of the unit circle. Understanding how to interpret this interval and identify all valid solutions within it is crucial for providing a complete answer.