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. 3 tan² x - tan x - 2 = 0

Verified step by step guidance

1. Recognize that the given equation is a quadratic in terms of \( \tan x \): \( 3 \tan^{2} x - \tan x - 2 = 0 \). Our goal is to solve for \( x \) in the interval \( [0, 2\pi) \).
2. Let \( t = \tan x \). Rewrite the equation as \( 3t^{2} - t - 2 = 0 \). This is a standard quadratic equation in \( t \).
3. Use the quadratic formula to solve for \( t \): \[ t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a = 3 \), \( b = -1 \), and \( c = -2 \).
4. Calculate the discriminant \( \Delta = b^{2} - 4ac \) and find the two possible values for \( t = \tan x \).
5. For each value of \( t \), solve \( \tan x = t \) on the interval \( [0, 2\pi) \). Recall that \( \tan x \) has period \( \pi \), so the solutions are \( x = \arctan(t) + k\pi \) for integers \( k \). Find the specific solutions within the given interval.

Key Concepts

Solving Quadratic Equations in Trigonometric Functions

This concept involves treating trigonometric equations like algebraic quadratics by substituting the trigonometric function (e.g., tan x) as a variable. The equation 3 tan² x - tan x - 2 = 0 can be solved using factoring or the quadratic formula to find values of tan x, which then lead to solutions for x.

Properties and Periodicity of the Tangent Function

Understanding that the tangent function has a period of π and is undefined at odd multiples of π/2 is crucial. Solutions for x must be found within the interval [0, 2π), considering the periodicity to identify all valid angles where tan x satisfies the equation.

Finding Exact and Approximate Solutions

After solving for tan x, one must find the corresponding angles x using inverse tangent functions. Exact values are preferred when possible, but approximate solutions to four decimal places are acceptable, especially when the inverse tangent yields irrational numbers.