Domanda del libro di testo
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos² x - cos x - 1 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 93
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0
Guida verificata passo dopo passo1
Recognize that the given equation is a quadratic in terms of \( \tan x \): \(4 \tan^{2} x - 8 \tan x + 3 = 0\). Our goal is to solve for \(x\) in the interval \([0, 2\pi)\).
Let \(t = \tan x\). Rewrite the equation as \(4t^{2} - 8t + 3 = 0\). 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 \(a=4\), \(b=-8\), and \(c=3\).
Calculate the two values of \(t\) from the quadratic formula. Each value corresponds to \(\tan x = t\).
For each value of \(t\), find the solutions for \(x\) by taking the arctangent: \[ x = \arctan(t) \]. Since \(\tan x\) is periodic with period \(\pi\), add \(\pi\) to each principal solution to find all solutions in \([0, 2\pi)\). Use a calculator to find the values of \(x\) correct to four decimal places.
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Solving Quadratic Equations in Trigonometric Functions
This involves treating the trigonometric expression, such as tan x, as a variable and solving the resulting quadratic equation using algebraic methods like factoring or the quadratic formula. The solutions for tan x are then used to find the corresponding angles x.
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Percorso guidato
6:24
Solving Quadratic Equations by Completing the Square
Using the Tangent Function and Its Properties
The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with period π, meaning solutions repeat every π radians. Understanding its domain and range helps identify valid solutions within the given interval.
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Percorso guidato
4:45How to Use a Calculator for Trig Functions
Finding Solutions on a Specified Interval with a Calculator
After solving for tan x, use a calculator to find the inverse tangent values and adjust for all solutions within the interval [0, 2π). Since tan x has period π, two solutions may exist per period. Round answers to the required decimal places as specified.
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Percorso guidato
3:17Inverse Tangent
Pratica correlata
Domanda del libro di testo
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 7 sin² x - 1 = 0
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Domanda del libro di testo
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. sin 2x + sin x = 0
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Domanda del libro di testo
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos x = ﹣ 2/5
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Domanda del libro di testo
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. 2 cos 2x + 1 = 0
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Domanda del libro di testo
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). tan x = ﹣3
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