Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problema 108

Capitolo 2, Problema 108

Concept Check Find a solution for each equation. sec(2θ + 6°) cos(5θ + 3°) = 1

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Recall the definition of secant: \(\sec x = \frac{1}{\cos x}\). Rewrite the equation \(\sec(2\theta + 6^\circ) \cos(5\theta + 3^\circ) = 1\) as \(\frac{1}{\cos(2\theta + 6^\circ)} \cdot \cos(5\theta + 3^\circ) = 1\).

Multiply both sides of the equation by \(\cos(2\theta + 6^\circ)\) to eliminate the fraction, giving \(\cos(5\theta + 3^\circ) = \cos(2\theta + 6^\circ)\).

Use the cosine equation property: if \(\cos A = \cos B\), then \(A = B + 360^\circ k\) or \(A = -B + 360^\circ k\), where \(k\) is any integer.

Set up the two equations based on the property: 1) \(5\theta + 3^\circ = 2\theta + 6^\circ + 360^\circ k\) 2) \(5\theta + 3^\circ = - (2\theta + 6^\circ) + 360^\circ k\)

Solve each linear equation for \(\theta\) by isolating \(\theta\) and expressing the general solution in terms of \(k\).

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Reciprocal Trigonometric Functions

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). Understanding this relationship allows us to rewrite or simplify equations involving secant and cosine, which is essential for solving the given equation.

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Introduction to Trigonometric Functions

Trigonometric Equation Solving

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the domain. This often requires using identities, inverse functions, and considering periodicity to find general solutions.

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How to Solve Linear Trigonometric Equations

Angle Sum and Multiple Angle Identities

Expressions like 2θ + 6° and 5θ + 3° involve linear combinations of the variable θ. Understanding how to handle these angles, including their periodic properties and how to solve equations with multiple angles, is crucial for finding all valid solutions.

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Double Angle Identities

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