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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 6.2.59
Capitolo 7, Problema 6.2.59

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
5 + 5 tan² θ = 6 sec θ

Guida verificata passo dopo passo
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Start by rewriting the given equation: \(5 + 5 \tan^{2} \theta = 6 \sec \theta\).
Recall the Pythagorean identity relating tangent and secant: \(1 + \tan^{2} \theta = \sec^{2} \theta\). Use this to express \(\tan^{2} \theta\) in terms of \(\sec^{2} \theta\) as \(\tan^{2} \theta = \sec^{2} \theta - 1\).
Substitute \(\tan^{2} \theta = \sec^{2} \theta - 1\) into the original equation to get \(5 + 5(\sec^{2} \theta - 1) = 6 \sec \theta\).
Simplify the equation to form a quadratic in \(\sec \theta\): \(5 + 5 \sec^{2} \theta - 5 = 6 \sec \theta\), which reduces to \(5 \sec^{2} \theta = 6 \sec \theta\).
Rewrite the equation as \(5 \sec^{2} \theta - 6 \sec \theta = 0\), factor it, and solve for \(\sec \theta\). Then, find the corresponding values of \(\theta\) by considering the definition \(\sec \theta = \frac{1}{\cos \theta}\) and solving for \(\theta\) in the specified domain, ensuring to express solutions in the least possible nonnegative angle measures.

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Trigonometric Identities

Trigonometric identities like tan²θ + 1 = sec²θ relate different trig functions and simplify equations. Recognizing and applying these identities helps transform the given equation into a solvable form by expressing all terms in a common function.
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Solving Trigonometric Equations

Solving trig equations involves isolating the trig function, finding general solutions using unit circle values, and applying periodicity to list all solutions. Understanding how to handle multiple solutions and restrictions on the domain is essential.
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Angle Measurement and Conversion

Angles can be measured in radians or degrees, and converting between them is often necessary. Solutions must be expressed in the least nonnegative angle measure, requiring knowledge of angle normalization and rounding conventions for approximate answers.
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Reference Angles on the Unit Circle
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