In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cos² x - 3 = 0

Verified step by step guidance

1. Start with the given equation: \(5 \cos^{2} x - 3 = 0\).
2. Isolate the cosine squared term by adding 3 to both sides: \(5 \cos^{2} x = 3\).
3. Divide both sides by 5 to solve for \(\cos^{2} x\): \(\cos^{2} x = \frac{3}{5}\).
4. Take the square root of both sides to solve for \(\cos x\): \(\cos x = \pm \sqrt{\frac{3}{5}}\).
5. Find all values of \(x\) in the interval \([0, 2\pi)\) where \(\cos x = \sqrt{\frac{3}{5}}\) and where \(\cos x = -\sqrt{\frac{3}{5}}\), using the unit circle or inverse cosine function.

Key Concepts

Solving Quadratic Trigonometric Equations

This involves treating trigonometric equations like algebraic quadratics by substituting expressions such as cos²x with a variable. After solving the quadratic equation, substitute back to find the trigonometric values and then solve for the angle x within the given interval.

Unit Circle and Interval Restrictions

Understanding the unit circle helps identify all angle solutions for trigonometric equations within a specified interval, here [0, 2π). Since cosine values repeat every 2π, solutions must be found within one full rotation, considering both positive and negative cosine values.

Exact Values and Approximate Solutions

Exact values refer to well-known trigonometric values expressed in terms of π or simple fractions, while approximate solutions use decimal values rounded to a specified precision. Knowing when and how to use each is essential for providing answers as required by the problem.