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 25

Capitolo 2, Problema 25

Concept Check What is wrong with the following item that appears on a trigonometry test? "Find sec θ , given that cos θ = 3/2 . "

Guida verificata passo dopo passo

Recall the definition of the cosine function: \(\cos \theta\) represents the ratio of the adjacent side to the hypotenuse in a right triangle, and its value must lie between -1 and 1 inclusive.

Examine the given value: \(\cos \theta = \frac{3}{2} = 1.5\), which is greater than 1.

Since \(\cos \theta\) cannot be greater than 1 or less than -1 for any real angle \(\theta\), the given value is not possible for any real angle.

Therefore, the problem is flawed because it asks to find \(\sec \theta\) based on an invalid cosine value.

To correct the problem, ensure that the given value of \(\cos \theta\) is within the valid range \([-1, 1]\) before asking to find \(\sec \theta\).

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Domain of the Cosine Function

The cosine of an angle in trigonometry represents the ratio of the adjacent side to the hypotenuse in a right triangle, and its value must lie between -1 and 1. A value like 3/2 (which is 1.5) is outside this range, making it impossible for cos θ to equal 3/2 in real numbers.

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Percorso guidato

3:43

Finding the Domain of an Equation

Definition of Secant Function

The secant function, sec θ, is defined as the reciprocal of the cosine function, i.e., sec θ = 1/cos θ. To find sec θ, the cosine value must be valid and non-zero; otherwise, sec θ is undefined or does not exist in the real number system.

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6:22

Graphs of Secant and Cosecant Functions

Validity of Trigonometric Values in Problems

When solving trigonometric problems, it is essential to verify that given values are within the permissible range for the function involved. Providing an invalid cosine value like 3/2 indicates a conceptual or typographical error, which must be identified before attempting to solve the problem.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (3 , ―4)

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Domanda del libro di testo

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (0, ―3)

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Domanda del libro di testo

Find the measure of each marked angle. See Example 2.

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Domanda del libro di testo

Find the measure of each marked angle. See Example 2.

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Domanda del libro di testo

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 147° 12' , 30° 19'