Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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63,186 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces the trigonometric functions, focusing on radian measure, the sine and cosine functions, and extending to differentiation, integration, and other trigonometric functions like tangent. A deep understanding of these concepts enables the description and analysis of cyclical and repetitive phenomena in nature, such as planetary motions and vibration patterns.

Learning Objectives

1

Explain the concept of radian measure of angles and its relation to degrees.

2

Describe the sine and cosine functions, including their periodic properties and applications.

3

Apply differentiation and integration techniques to sin(t) and cos(t).

4

Analyze the tangent and other trigonometric functions and understand their role in modeling cyclical phenomena.

5

Interpret real-world phenomena using trigonometric functions to describe repetitive processes.

Key Concepts

CONCEPT

DEFINITION

Radian Measure

A way of measuring angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius.

Sine Function

A trigonometric function that represents the y-coordinate of a point on the unit circle, characterized by its periodic behavior.

Cosine Function

A trigonometric function that represents the x-coordinate of a point on the unit circle, periodic and phase-shifted relative to the sine function.

Periodic Function

A function that repeats its values in regular intervals or periods, such as sine and cosine functions.

Differentiation of Trigonometric Functions

The process of finding the derivative of trigonometric functions, e.g., derivative of sin(t) is cos(t) and derivative of cos(t) is -sin(t).

Integration of Trigonometric Functions

The reverse process of differentiation, used to find the antiderivative of trigonometric functions.

Tangent Function

A trigonometric function defined as sin(t)/cos(t), with its own periodic behavior and vertical asymptotes.

Example Problems

Example 1

Convert the following to radian measure. $$30^{\circ}, 120^{\circ}, 315^{\circ}$$

Example 2

Convert the following to radian measure. $$18^{\circ}, 72^{\circ}, 150^{\circ}$$

Example 3

Convert the following to radian measure. $$990^{\circ},-270^{\circ},-540^{\circ}$$

Example 4

Convert the following to radian measure. $$450^{\circ},-210^{\circ},-90^{\circ}$$

Example 5

Give the radian measure of each angle described.
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Step-by-Step Explanations

QUESTION

How do you convert an angle from degrees to radians?

STEP-BY-STEP ANSWER:

Step 1: Recall the conversion factor: 180° equals π radians.
Step 2: Multiply the degree measure by π/180.
Step 3: Simplify the expression to obtain the radian measure.
Final Answer: For an angle of d degrees, the radian measure is (d × π/180).

Radian Measure

QUESTION

What is the derivative of sin(t) with respect to t?

STEP-BY-STEP ANSWER:

Step 1: Recall the basic differentiation rule for sine: (d/dt) sin(t) = cos(t).
Step 2: Apply the differentiation directly to sin(t).
Final Answer: The derivative of sin(t) is cos(t).

Differentiation of sin(t)

QUESTION

How do you integrate cos(t) with respect to t?

STEP-BY-STEP ANSWER:

Step 1: Recall the basic integration rule for cosine: ∫cos(t) dt = sin(t) + C.
Step 2: Set up the integral ∫cos(t) dt.
Step 3: Use the antiderivative formula for cos(t).
Final Answer: ∫cos(t) dt = sin(t) + C.

Integration of cos(t)

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Common Mistakes

  • Confusing degrees with radians when converting angle measures.
  • Overlooking the periodic nature of sine and cosine functions, leading to misinterpretation of their graphs.
  • Forgetting the constant of integration when integrating trigonometric functions.
  • Applying differentiation rules incorrectly by not considering negative signs, especially in differentiating cosine function.