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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
Capitolo 3, Problema 108

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. tan x sec x = 2 tan x

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Start with the given equation: \(\tan x \sec x = 2 \tan x\).
Bring all terms to one side to set the equation to zero: \(\tan x \sec x - 2 \tan x = 0\).
Factor out the common factor \(\tan x\): \(\tan x (\sec x - 2) = 0\).
Set each factor equal to zero and solve separately: 1) \(\tan x = 0\) 2) \(\sec x - 2 = 0\).
For \(\tan x = 0\), find all \(x\) in \([0, 2\pi)\) where tangent is zero. For \(\sec x - 2 = 0\), rewrite as \(\sec x = 2\), then use the identity \(\sec x = \frac{1}{\cos x}\) to find \(\cos x = \frac{1}{2}\) and solve for \(x\) in \([0, 2\pi)\).

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Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing identities like sec x = 1/cos x and the relationship between tan x and sin x/cos x helps simplify and solve the equation efficiently.
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