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. tan x = 2 cos x tan x

Verified step by step guidance

1. Start by writing down the given equation: \(\tan x = 2 \cos x \tan x\).
2. Recall that \(\tan x = \frac{\sin x}{\cos x}\), so substitute this into the equation to express everything in terms of sine and cosine: \(\frac{\sin x}{\cos x} = 2 \cos x \cdot \frac{\sin x}{\cos x}\).
3. Simplify the right side by canceling \(\cos x\) where possible, keeping in mind the domain restrictions where \(\cos x \neq 0\) to avoid division by zero.
4. Rearrange the equation to isolate terms and set it equal to zero, which will allow factoring or using trigonometric identities to find solutions.
5. Solve the resulting equation(s) for \(x\) within the interval \([0, 2\pi)\), considering all possible cases including when \(\cos x = 0\) (since division by zero was excluded earlier).

Key Concepts

Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. Solving these equations means finding all angle values within a specified interval that satisfy the equation. Understanding how to manipulate and simplify these equations is essential for finding exact or approximate solutions.

Interval Notation and Domain Restrictions

The problem restricts solutions to the interval [0, 2π), meaning all solutions must be found between 0 and just before 2π radians. Recognizing this domain helps limit the possible solutions and ensures answers are relevant to the given range.

Relationship Between Tangent and Cosine Functions

The equation involves both tangent and cosine functions, which are related through sine and cosine (tan x = sin x / cos x). Understanding how to express tangent in terms of sine and cosine allows for algebraic manipulation and simplification, facilitating the solving process.