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. cos x - 5 = 3 cos x + 6

Verified step by step guidance

1. Start by writing down the given equation: \(\cos x - 5 = 3 \cos x + 6\).
2. Group like terms to isolate the cosine terms on one side. Subtract \(\cos x\) from both sides and subtract 6 from both sides to get: \(-5 - 6 = 3 \cos x - \cos x\).
3. Simplify both sides: \(-11 = 2 \cos x\).
4. Solve for \(\cos x\) by dividing both sides by 2: \(\cos x = \frac{-11}{2}\).
5. Analyze the value of \(\cos x = -\frac{11}{2}\). Since cosine values must be between -1 and 1, this equation has no solution on the interval \([0, 2\pi)\).

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given interval. This often requires algebraic manipulation and applying inverse trigonometric functions to determine exact or approximate angle values.

Interval Restriction [0, 2π)

The interval [0, 2π) represents one full rotation around the unit circle, covering all possible angle measures in radians for one cycle of trigonometric functions. Solutions must be found only within this range, ensuring all answers correspond to angles between 0 (inclusive) and 2π (exclusive).

Using Exact and Approximate Values

Exact values refer to well-known trigonometric values expressed in terms of fractions of π or radicals, while approximate values are decimal representations rounded to a specified precision. Understanding when to use each helps provide precise or practical solutions depending on the problem's requirements.