Mathematics 101 Transitional Mathematics

Study Guide :: Unit 6

Trigonometry: As the Geometry of Angles in the Plane

Ways and means of measuring angles began in antiquity.

The rotational measure of angles can be traced to various ancient civilizations in the geographical regions of Mesopotamia (between the Tigris and Euphrates rivers), Egypt, and the Indus River basin (located in Pakistan). However, it is the Babylonians (circa 1700 B.C.E.) who are credited with the introduction of the degree unit of rotational measurement. They were the ones who subdivided a full rotation into 360 angular parts and called each of them an angle of one degree. The division was not arbitrary. It was derived using their number system, which had a base of 60. (In contrast, our decimal system has a base of 10.)

This type of angular measurement was adopted by various cultures in that region and over time made its way to the farthest reaching climes of the Roman Empire, which extended to its northern edge, the Antonine Wall, located in what is now central Scotland. It is rather interesting to note that very close (relatively speaking) to that most northern frontier, approximately 3,400 years later, an English mathematician, Roger Cotes (1682–1716) added a new dimension to the measurement of angles. He introduced the concept of, what is now termed, the radian measure of angles, though he did not coin the nomenclature itself. The advantage of radian measure was that it was an arc length along a circle and, therefore, could be regarded as a linear measure of an angle because the arc could be flattened along a horizontal axis. This revolutionized the study of the trigonometric ratios and led to a visual depiction of their graphs in the Cartesian plane.

Learning Objectives

After you have completed Unit 6, you should be able to:

1. compute three basic ways of measuring angles—in degrees, in radians, and as trigonometric ratios; namely the sine, the cosine, the tangent, the cosecant, the secant and the cotangent of a given angle;
2. explain the relationship between degrees and radians and convert from one to the other;
3. compute the length of a circular arc and the area of a circular sector;
4. define and calculate Exact Values (not the approximated values obtained from a scientific calculator) of the six trigonometric ratios for the special angles 30°, 45°, 60°, the axis angles, and any angles coterminal with them;
5. define the six trigonometric ratios as functions of a single variable;
6. list the fundamental identities (five reciprocal and three Pythagorean) associated with the six trigonometric functions;
7. know the statements of the Law of Sines and the Law of Cosines and define their role in finding areas and angles in triangular forms;
8. calculate the area of any triangle knowing either the lengths of two of its sides and the angle between them or the lengths of its three sides; and
9. apply trigonometric functions to real-world situations.

Readings and Practice

Chapter 6 (pp. 229–284) of the textbook

Optional Sections

• Inverse Trigonometric Functions and Right Triangles (pp. 258–265)

  If you are preparing for a Calculus course, it is good to know how to define and evaluate the inverse trigonometric functions. You should also be aware that evaluating an inverse trigonometric function is equivalent to solving a corresponding trigonometric equation. For example, the value of the inverse sine function at the number ${\displaystyle 1/2}$ is the solution angle ${\displaystyle \theta }$ of the equation ${\displaystyle \sin \theta =1/2}$. Because the latter equation has literally an infinity of solutions, we choose the one which is in the range of the inverse sine function. This is expressed as either

  ${\displaystyle \theta ={\sin }^{-1}(1/2)=\pi /6}$   or   ${\displaystyle \theta =\arcsin (1/2)=\pi /6}$.

  The latter nomenclature seems rather odd but the ${\displaystyle \arcsin (1/2)}$ is the arc length on the unit circle subtended by the angle ${\displaystyle \theta }$ whose sine value is ${\displaystyle 1/2}$. That, of course, is the measure of that angle ${\displaystyle \theta }$ in radians.

• Focus on Modeling: Surveying (pp. 285–288)

Online graphing tool: Graph

Angle Measure

Textbook Readings

pp. 229–234

Optional Subsection

• Circular Motion (pp. 234–235)

Practice

(pp. 236–239):  1, 2, 3, 5, 9, 13, 15, 17, 19, 27, 29, 35, 37, 41, 43, 45, 47, 51, 53, 57, 61, 73, 87

Answers

p. 516

Terms to Understand

formal definition of radian measure of angles and how it is related to degree measure; standard position of an angle in the Cartesian plane; coterminal angles in the plane; length of a circular arc; area of a circular sector

Figures/Tables

Definition of Radian Measure (p. 230)
Relationship between Degrees and Radians (p. 231)
Length of a Circular Arc (p. 233)
Area of a Circular Sector (p. 234)

Online Tutorials—the Overview

Pertinent to “Angle Measure”

• Measurement of Angles: degree to radian measure and back again

Online Maple TA Practice

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

Trigonometry of Right Triangles

In this section, we are talking about the trigonometry of positive acute angles (angles between 0° and 90°) and their exact values (not their approximated values using a scientific calculator).

Textbook Readings

pp. 239–243

Practice

(pp. 244–247):  1, 2, 3, 5, 9, 13, 17, 19, 21, 25, 29, 31, 35, 41, 47, 49, 53, 61

Answers

pp. 516–517

Terms to Understand

sine, cosine, tangent, cosecant, secant, and cotangent of a positive acute angle as a ratio of sides in a right triangle

Figures/Tables

The Trigonometric Ratios (p. 239)
Trigonometric Ratios for Special Acute Angles (p. 241)

Online Tutorials—the Overview

Pertinent to “Trigonometry of Right Triangles”

• Measurement of Angles: trigonometric measure
• Descartes’ Coordinate Plane
• The Trigonometry of Angles: quadrant I in the plane

Online Maple TA Practice

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

Trigonometric Functions of Angles

Note that, when referring to ‘functions,’ we allow the angle to vary and take on arbitrary radian values of any size, both positive and negative.

Textbook Readings

pp. 247–255

Practice

(pp. 255–258):  1, 2, 3, 7, 13, 15, 21, 23, 27, 31, 33, 35–38, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 67

Note that, in questions 45, 47, 49, and 51, you are solving a trigonometric equation with the unknown being the angle ${\displaystyle \theta }$.

Answers

p. 517

Terms to Understand

sine, cosine, tangent, cosecant, secant, and cotangent of any angle using the CAST formula; definition of the six trigonometric functions as functions of a real variable; reference (or related angle to a given angle in standard position; trigonometric identity

Figures/Tables

Definition of the Trigonometric Functions (p. 248)
Relationship between Trigonometric Functions and Real Numbers (p. 249)

(how degree measure of angles can be interpreted as linear measure on the real line, leading to the trigonometric ratios as functions of a real variable)

Signs of the Trigonometric Functions (CAST formula) (p. 250)

Reference (or related) Angle (p. 250)

Evaluating Trigonometric Functions for Any Angle (p. 251)

Fundamental Identities (p. 253)

Area of a Triangle (expressed as a trigonometric function) (p. 255)

Online Tutorials—the Overview

Pertinent to “Trigonometric Functions of Angles”

• The Trigonometry of Angles: quadrants I to IV in the plane
  Quadrant I Angles
  Quadrant II Angles
  Quadrant III Angles
  Quadrant IV Angles
• The Trigonometry of Angles: the axis angles
• The Trigonometry of Angles: negative angles
• Slope of a Line: as the tangent measure of the line’s angle of inclination

Online Maple TA Practice

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

Law of Sines

Textbook Readings

pp. 265–269

Practice

(pp. 269–271):  1, 2, 3, 7, 11, 15, 19, 23, 29, 33, 35, 41

Answers

p. 517

Terms to Understand

Law of Sines

Figures/Tables

The Law of Sines (p. 265)

Online Tutorials—the Overview

Pertinent to “Law of Sines”

• None

Online Maple TA Practice

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

Law of Cosines

Textbook Readings

pp. 272–276

Practice

(pp. 276–279):  1, 2, 3, 9, 11, 15, 17, 21, 27, 31, 33, 39, 45

Answers

pp. 517–518

Terms to Understand

Law of Cosines (p. 272)

Figures/Tables

The Law of Cosines (p. 272)
Heron’s Formula (p. 275)

Online Tutorials—the Overview

Pertinent to “Law of Cosines”

• None

Online Maple TA Practice

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

Unit 6 Review

(pp. 279–282): odd numbers

Answers

p. 518

Unit 6 Test

(pp. 283–284): 1–12

Answers

p. 518

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)