Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 25

Chapter 2, Problem 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 . "

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

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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Textbook Question

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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Textbook Question

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

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Textbook Question

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

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Textbook Question

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