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 7

Chapter 3, Problem 7

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, such as the sides of the right triangle or the coordinates on the unit circle, since sin A, cos A, and tan A depend on these values.

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 sides of the triangle are not given, use the Pythagorean theorem \(a^2 + b^2 = c^2\) to find the missing side lengths, where \(c\) is the hypotenuse.

Substitute the known side lengths into the formulas for \(\sin A\), \(\cos A\), and \(\tan A\) to write expressions for each function.

Simplify the expressions if possible, and express the values exactly (e.g., in terms of square roots or fractions) rather than decimal approximations.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Definition of Sine, Cosine, and Tangent

Sine, cosine, and tangent are fundamental trigonometric functions relating 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.

Using Reference Triangles and Special Angles

Reference triangles, especially those with angles of 30°, 45°, and 60°, provide exact trigonometric values. Recognizing these special angles and their side ratios allows for precise calculation of sin A, cos A, and tan A without a calculator.

Trigonometric Identities and Relationships

Key identities like tan A = sin A / cos A and the Pythagorean identity sin² A + cos² A = 1 help relate the functions and verify results. These relationships are useful for expressing one function in terms of others and simplifying expressions.

Recommended video:

5:32

Fundamental Trigonometric Identities

Related Practice

Textbook Question

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.

cot 30°

views

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. -135° C. 82° 8. D. 30° 9. E. 38° 10. F. 32°

views

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. 212° B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. F. 32°

views

Textbook Question

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

Column I: 1.

sec 18°

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

views

Textbook Question

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

Column I: 1.

scs 80°

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

views

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°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

views