Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경

Lial 12th Edition

Ch. 1 - Trigonometric Functions

문제 71

2장, 문제 71

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

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Recall the identity relating cosecant and sine: \(\csc \theta = \frac{1}{\sin \theta}\). Use this to find \(\sin \theta\) by taking the reciprocal of \(\csc \theta\).

Since \(\csc \theta = -1.45\), calculate \(\sin \theta = \frac{1}{-1.45}\). Keep this as a fraction or decimal for now without simplifying fully.

Determine the sign of \(\cos \theta\) in quadrant III. In quadrant III, both sine and cosine are negative, so \(\cos \theta < 0\).

Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\). Substitute the value of \(\sin \theta\) and solve for \(\cos \theta\), taking the negative root because of the quadrant.

Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Substitute the values of \(\cos \theta\) and \(\sin \theta\) found in previous steps to express \(\cot \theta\). Then rationalize the denominator if necessary.

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주요 개념

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

Reciprocal Trigonometric Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as csc θ = 1/sin θ and cot θ = 1/tan θ. Knowing csc θ allows you to find sin θ, which is essential for determining cot θ.

Sign of Trigonometric Functions in Quadrants

The sign of trig functions depends on the quadrant of the angle. In quadrant III, both sine and cosine are negative, while tangent and cotangent are positive. This helps determine the correct sign of cot θ.

Pythagorean Identity and Rationalizing Denominators

The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of cosine once sine is known. Rationalizing denominators ensures the final answer is simplified and free of radicals in the denominator, as required.

관련 실천

교과서 질문

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. -25.485°

707

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교과서 질문

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7. tan θ = ―15/8 , and θ is in quadrant II .

1021

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교과서 질문

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

672

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교과서 질문

Find the indicated function value. If it is undefined, say so. See Example 4. tan 450°

640

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교과서 질문

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. 59.0854°

707

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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.)

804

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