Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경

Lial 12th Edition

Ch. 1 - Trigonometric Functions

문제 67

2장, 문제 67

Find the indicated function value. If it is undefined, say so. See Example 4. sec 180°

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Recall the definition of the secant function: \(\sec \theta = \frac{1}{\cos \theta}\).

Identify the angle given: \(180^\circ\).

Find the cosine of \(180^\circ\): \(\cos 180^\circ\).

Evaluate \(\cos 180^\circ\) using the unit circle or known values. (Note: \(\cos 180^\circ = -1\).)

Calculate \(\sec 180^\circ\) by taking the reciprocal of \(\cos 180^\circ\), i.e., \(\sec 180^\circ = \frac{1}{\cos 180^\circ}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). It is defined wherever cosine is not zero. Understanding this reciprocal relationship is essential to evaluate secant values for given angles.

Cosine of Special Angles

Cosine values for special angles like 0°, 90°, 180°, and 270° are fundamental. For 180°, cos(180°) = -1. Knowing these values helps directly compute sec(180°) by taking the reciprocal of cosine at that angle.

Domain and Undefined Values of Trigonometric Functions

Trigonometric functions can be undefined when their denominators are zero. Since sec(θ) = 1/cos(θ), sec(θ) is undefined where cos(θ) = 0. Recognizing these points prevents errors in evaluation and helps identify when a function value does not exist.

관련 실천

교과서 질문

Determine whether each statement is possible or impossible. See Example 4. cot θ = ―6

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교과서 질문

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). 174.255°

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교과서 질문

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.

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교과서 질문

In each figure, there are two similar triangles. Find the unknown measurement. Give approximations to the nearest tenth.

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교과서 질문

Find the indicated function value. If it is undefined, say so. See Example 4. sin(―270°)

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교과서 질문

Solve each problem. Solar Eclipse on Earth The sun has a diameter of about 865,000 mi with a maximum distance from Earth's surface of about 94,500,000 mi. The moon has a smaller diameter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moon's umbra (shadow) to reach the surface of Earth. (Data from Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners, Editors, Fundamental Astronomy, Fourth Edition, Springer-Verlag.) a. Calculate the maximum distance, to the nearest thousand miles, that the moon can be from Earth and still have a total solar eclipse occur. (Hint: Use similar triangles.)

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