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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 116
Capitolo 3, Problema 116

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

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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 \( [0, 2\pi) \).
Let \( t = \tan x \). Rewrite the equation as \( 3t^{2} - t - 2 = 0 \). This is a standard quadratic equation in \( t \).
Use the quadratic formula to solve for \( t \): \[ t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a = 3 \), \( b = -1 \), and \( c = -2 \).
Calculate the discriminant \( \Delta = b^{2} - 4ac \) and find the two possible values for \( t = \tan x \).
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.

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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.
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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.
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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.
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