Ch. 2 - Acute Angles and Right Triangles

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

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 9

Chapter 3, Problem 9

Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

Verified step by step guidance

Identify the given information about angle A from the problem or diagram, such as the lengths of sides in a right triangle or the coordinates on the unit circle.

Recall the definitions of the trigonometric functions in a right triangle: \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\).

If the problem involves a right triangle, use the Pythagorean theorem \(a^2 + b^2 = c^2\) to find any missing side lengths needed to compute the ratios for sine, cosine, and tangent.

Substitute the known side lengths into the definitions to write expressions for \(\sin A\), \(\cos A\), and \(\tan A\) in terms of these lengths.

Simplify the expressions if possible, such as reducing fractions or rationalizing denominators, to find the exact values or expressions for \(\sin A\), \(\cos A\), and \(\tan A\).

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Key Concepts

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

Trigonometric Ratios

Trigonometric ratios—sine, cosine, and tangent—relate the angles of a right triangle to the ratios of its sides. Specifically, sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, and tan A = opposite/adjacent. Understanding these definitions is essential for finding exact values of these functions.

Exact Values of Common Angles

Certain angles like 0°, 30°, 45°, 60°, and 90° have well-known exact trigonometric values derived from special triangles or the unit circle. Recognizing these angles and their sine, cosine, and tangent values allows for precise calculation without a calculator.

Reference to Example Problems

Using example problems helps illustrate the method to find sin A, cos A, and tan A, often involving drawing triangles, applying definitions, or using identities. Reviewing Example 1 provides a step-by-step approach to solving similar trigonometric questions.

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Reference Angles on the Unit Circle

Related Practice

Textbook Question

Determine whether each statement is true or false. If false, tell why. csc 22° ≤ csc 68°

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Textbook Question

Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. 480° F. 32°

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Textbook Question

Concept Check Refer to the discussion of accuracy and significant digits in this section to answer the following. Mt. Everest When Mt. Everest was first surveyed, the surveyors obtained a height of 29,000 ft to the nearest foot. State the range represented by this number. (The surveyors thought no one would believe a measurement of 29,000 ft, so they reported it as 29,002.) (Data from Dunham, W., The Mathematical Universe, John Wiley and Sons.)

Textbook Question

Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

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Textbook Question

Determine whether each statement is true or false. If false, tell why. tan 60° ≥ cot 40°

Textbook Question

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

tan⁻¹ 30

Column II:

A. 88.09084757°

B. 63.25631605°