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. insert step 1: Start by simplifying the given equation $\tan x=2\cos x\tan x$.
2. insert step 2: Factor out $\tan x$ from both sides of the equation: $\tan x(1-2\cos x)=0$.
3. insert step 3: Set each factor equal to zero: $\tan x=0$ and $1-2\cos x=0$.
4. insert step 4: Solve $\tan x=0$ for $x$ in the interval $[0,2\pi )$.
5. insert step 5: Solve $1-2\cos x=0$ for $x$ in the interval $[0,2\pi )$.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, reciprocal identities, and quotient identities. Understanding these identities is crucial for simplifying trigonometric equations and solving for unknown angles.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy a given trigonometric equation. This often requires manipulating the equation using identities, isolating the trigonometric function, and determining the general solutions. Solutions are typically expressed in terms of radians, especially when working within specific intervals like [0, 2π).

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [0, 2π) indicates that the solutions must be within the range starting from 0 (inclusive) to 2π (exclusive). Understanding how to interpret and work within these intervals is essential for correctly identifying valid solutions to trigonometric equations.