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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.2.37
Capitolo 7, Problema 6.2.37

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sec² θ tan θ = 2 tan θ

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Start by writing down the given equation: \(\sec^{2} \theta \tan \theta = 2 \tan \theta\).
Recognize that \(\tan \theta\) is a common factor on both sides. To simplify, subtract \(2 \tan \theta\) from both sides to get: \(\sec^{2} \theta \tan \theta - 2 \tan \theta = 0\).
Factor out \(\tan \theta\) from the left side: \(\tan \theta (\sec^{2} \theta - 2) = 0\).
Set each factor equal to zero to find possible solutions: (1) \(\tan \theta = 0\) and (2) \(\sec^{2} \theta - 2 = 0\).
For (2), rewrite \(\sec^{2} \theta\) in terms of \(\tan^{2} \theta\) using the identity \(\sec^{2} \theta = 1 + \tan^{2} \theta\), then solve for \(\tan \theta\). Finally, find all \(\theta\) in \([0^\circ, 360^\circ)\) that satisfy these conditions.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that sec²θ = 1 + tan²θ helps simplify and solve the equation by expressing all terms in a single trigonometric function.
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Fundamental Trigonometric Identities

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This often requires factoring, using identities, and considering the periodic nature of functions like tangent and secant.
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How to Solve Linear Trigonometric Equations

Interval and Solution Set for Angles

When solving trigonometric equations over [0°, 360°), it is essential to find all solutions within one full rotation of the unit circle. Understanding how to interpret and express solutions as exact values or decimal approximations ensures completeness and accuracy.
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Coterminal Angles
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