Mathematics 101 Transitional Mathematics

Study Guide :: Unit 7

The Trigonometric Functions: Exemplars of Periodic Motion

When the radian measure of angles was introduced in the early eighteenth century by Roger Cotes and the trigonometric ratios of angles in the plane began to be viewed as functions of a linearized angle, the undulating movement of the sine and cosine graphs fed the imaginations of scientists working in areas as diverse as the swinging motion of pendula, the vibration of musical notes, the ebb and flow of tides, electromagnetic radiation waves, the modeling of alternating electrical currents, and other physical phenomena on or near the earth’s surface.

What is truly phenomenal is that even quite general functions, which may not necessarily be strictly harmonic in nature, can be represented and approximated (under certain conditions) by sums of relatively simple trigonometric functions. The study of the ways this can happen is called ‘Fourier Analysis,’ named after the French mathematician and physicist, Joseph Fourier (1768–1830), who showed that representing a function as a sum of trigonometric functions greatly simplified the study of the propagation of heat.

Today, Fourier Analysis, with its foundational roots in the six basic trigonometric functions, has numerous scientific applications—in physics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, sound navigation and ranging (sonar), optics, diffraction, and protein structure analysis, to mention only some of them.

Learning Objectives

After you have completed Unit 7, you should be able to:

explain the relationship between degrees and radians and convert from one to the other;
define and calculate the exact values of the six trigonometric ratios for the special angles 0, π/6, π/4, π/3 and π/2 radians, and any angles coterminal with them without a calculator;
define the six trigonometric ratios as functions of a single variable expressed in radians;
ist the domains, ranges, and periodicity of each of the six trigonometric functions;
modify each of the six trigonometric functions by both rigid and non-rigid transformations and interpret their meanings in terms of domain, ranges, and periodicity; and
apply, when appropriate, the trigonometric functions to real-world situations.

Readings and Practice

Chapter 7 (pp. 289–326, 332–343) of the textbook

Optional Sections

The Unit Circle (pp. 290–297)
Inverse Trigonometric Functions and Their Graphs (pp. 326–332)
Fitting Sinusoidal Curves to Data (pp. 347–352)

Online graphing tool: Graph

Trigonometric Functions of Realnumbers

This section is a review of Unit 6 except that the angles are measured in radians along the line of real numbers, rather than in degrees as angles in a right triangle.

Recall that an angle $θ$ in the plane has a radian measure of ‘ $t$ ’ = the length of the arc along the unit circle (circle of radius 1) subtended by that angle $θ$ in its standard position. The terminal arm of an angle $θ$ of $t$ radians intersects the unit circle at the point $P = (x, y) = (\cos θ, \sin θ)$ .

Also note that the term ‘reference number’ in this section of the text is simply the reference/related angle measured in radians.

Textbook Readings

pp. 297–304

Practice

(pp. 304–306): 1, 2, 3, 4, 5, 9, 13, 15, 19, 23, 27, 29, 33, 37, 39, 43, 47, 53, 55, 57, 65, 69, 71, 73, 75, 81, 85 for one of the reduction formulas given

Answers

pp. 518–519

Terms to Understand

domain of each of the six trigonometric functions as functions of a real variable

Figures/Tables

Definition of the Trigonometric Functions (p. 297)
Trigonometric Ratios for Special Acute Angles (p. 298)

Relationship between Trigonometric Functions and Degree Measure (p. 299)

Domains of the Trigonometric Functions (p. 300)
Signs of the Trigonometric Functions (CAST formula) (p. 300)
Even-Odd Properties of the Trigonometric Functions (p. 302)
Fundamental Identities (p. 302)

Online Tutorials—the Overview

Pertinent to “Trigonometric Functions of Realnumbers”

The Trigonometric Ratios as Functions: an introduction
The Sine Function: a closer look
How to Derive or Prove Some Trigonometric Identities: Part I
How to Derive or Prove Some Trigonometric Identities: Part II

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Trigonometric Graphs

Textbook Readings

pp. 306–315

Practice

(pp. 316–318): 1, 2, 3, 5, 11, 15, 19, 21, 25, 27, 29, 33, 37, 41, 43, 45, 47, 49, 61, 71, 75, 77

For 3, 5, 11, and 15, try graphing them by using transformations rather than a graphing device.

Answers

pp. 519–520

Terms to Understand

period, amplitude, and phase shift of a sine or cosine function

Figures/Tables

Periodic Properties of Sine and Cosine (p. 307)
Sine and Cosine Curves (p. 310)
Shifted Sine and Cosine Curves (p. 311)

Online Tutorials—the Overview

Pertinent to “Trigonometric Graphs”

Plotting the graph of the sine function
Non-rigid transformations of functions: expansions and contractions—corresponds to the changing periods and amplitudes of the sine and cosine curves
Rigid transformations of functions: translations and reflections—corresponds to the various phase shifts of the sine and cosine curves

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

More Trigonometric Graphs

Textbook Readings

pp. 319–324

Practice

(pp. 325–326): 1, 2, 3, 5, 7, 9, 13, 17, 21, 25, 27, 31, 35, 39, 41, 45, 53, 57

Try graphing the trigonometric functions by using transformations; then check your answers using a graphing device.

Answers

pp. 520–522

Terms to Understand

period, amplitude, and phase shift of a tangent, cotangent, secant, and cosecant function

Figures/Tables

Periodic Properties of Tangent, Cotangent, Secant, and Cosecant (p. 319)
Tangent and Cotangent Curves (p. 322)
Cosecant and Secant Curves (p. 323)

Online Tutorials—the Overview

Pertinent to “More Trigonometric Graphs”

None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Trigonometric Functions’ topic page.

Modeling Harmonic Motion

This is a direct application of the periodic nature of the sine and cosine functions.

Textbook Readings

pp. 332–338

Optional Subsection

Damped Harmonic Motion (pp. 338–340)

Practice

(pp. 340–343): 1, 2, 3, 7, 11, 15, 27

Answers

pp. 522–523

Terms to Understand

simple harmonic motion with its notions of amplitude, period, and frequency

Figures/Tables

Simple Harmonic Motion (p. 333)

Online Tutorials—the Overview

Pertinent to “Modeling Harmonic Motion”

None

Unit 7 Review

(pp. 343–345): odd numbers

Answers

pp. 523–524

Unit 7 Test

(p. 346): 1–9, 11–13 only

Answers

p. 524

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text and pp. 569–570 at the end of the text)