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TRIGONOMETRY

MICHAEL CORRAL

Trigonometry

Michael Corral

Schoolcraft College

About the author:

Michael Corral is an Adjunct Faculty member of the Department of Mathematics at

Schoolcraft College. He received a B.A. in Mathematics from the University of California

at Berkeley, and received an M.A. in Mathematics and an M.S. in Industrial & Operations

Engineering from the University of Michigan.

This text was typeset in LATEX with the KOMA-Script bundle, using the GNU Emacs

text editor on a Fedora Linux system. The graphics were created using TikZ and Gnuplot.

Copyright © 2009 Michael Corral.

Permission is granted to copy, distribute and/or modify this document under the terms of the

GNU Free Documentation License, Version 1.3 or any later version published by the Free

Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover

Texts. A copy of the license is included in the section entitled “GNU Free Documentation

License.”

Preface

This book covers elementary trigonometry. It is suitable for a one-semester course at the

college level, though it could also be used in high schools. The prerequisites are high school

algebra and geometry.

This book basically consists of my lecture notes from teaching trigonometry at Schoolcraft

College over several years, expanded with some exercises. There are exercises at the end

of each section. I have tried to include some more challenging problems, with hints when

I felt those were needed. An average student should be able to do most of the exercises.

Answers and hints to many of the odd-numbered and some of the even-numbered exercises

are provided in Appendix A.

This text probably has a more geometric feel to it than most current trigonometry texts.

That was, in fact, one of the reasons I wanted to write this book. I think that approaching the

subject with too much of an analytic emphasis is a bit confusing to students. It makes much

of the material appear unmotivated. This book starts with the “old-fashioned” right triangle

approach to the trigonometric functions, which is more intuitive for students to grasp.

In my experience, presenting the definitions of the trigonometric functions and then im-

mediately jumping into proving identities is too much of a detour from geometry to analysis

for most students. So this book presents material in a very different order than most books

today. For example, after starting with the right triangle definitions and some applications,

general (oblique) triangles are presented. That seems like a more natural progression of

topics, instead of leaving general triangles until the end as is usually the case.

The goal of this book is a bit different, too. Instead of taking the (doomed) approach that

students have to be shown that trigonometry is “relevant to their everyday lives” (which

inevitably comes off as artificial), this book has a different mindset: preparing students

to use trigonometry as it is used in other courses. Virtually no students will ever in their

“everyday life” figure out the height of a tree with a protractor or determine the angular

speed of a Ferris wheel. Students are far more likely to need trigonometry in other courses

(e.g. engineering, physics). I think that math instructors have a duty to prepare students

for that.

In Chapter 5 students are asked to use the free open-source software Gnuplot to graph

some functions. However, any program can be used for those exercises, as long as it produces

accurate graphs. Appendix B contains a brief tutorial on Gnuplot.

There are a few exercises that require the student to write his or her own computer pro-

gram to solve some numerical computation problems. There are a few code samples in Chap-

ter 6, written in the Java and Python programming languages, hopefully sufficiently clear

so that the reader can figure out what is being done even without knowing those languages.

iii

iv

PREFACE

Octave and Sage are also mentioned. This book probably discusses numerical issues more

than most texts at this level (e.g. the numerical instability of Heron’s formula for the area

of a triangle, the secant method for solving trigonometric equations). Numerical methods

probably should have been emphasized even more in the text, since it is rare when even a

moderately complicated trigonometric equation can be solved with elementary methods, and

since mathematical software is so readily available.

I wanted to keep this book as brief as possible. Someone once joked that trigonometry

is two weeks of material spread out over a full semester, and I think that there is some

truth to that. However, some decisions had to be made on what material to leave out. I had

planned to include sections on vectors, spherical trigonometry - a subject which has basically

vanished from trigonometry texts in the last few decades (why?) - and a few other topics,

but decided against it. The hardest decision was to exclude Paul Rider’s clever geometric

proof of the Law of Tangents without using any sum-to-product identities, though I do give

a reference to it.

This book is released under the GNU Free Documentation License (GFDL), which allows

others to not only copy and distribute the book but also to modify it. For more details, see

the included copy of the GFDL. So that there is no ambiguity on this matter, anyone can

make as many copies of this book as desired and distribute it as desired, without needing

my permission. The PDF version will always be freely available to the public at no cost (go

to http://www.mecmath.net/trig). Feel free to contact me at mcorral@schoolcraft.edu for any questions on this or any other matter involving the book (e.g. comments, suggestions,

corrections, etc). I welcome your input.

July 2009

MICHAEL CORRAL

Livonia, Michigan

Contents

Preface

iii

1

Right Triangle Trigonometry

1

1.1

Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Trigonometric Functions of an Acute Angle . . . . . . . . . . . . . . . . . . . .

7

1.3

Applications and Solving Right Triangles . . . . . . . . . . . . . . . . . . . . . 14

1.4

Trigonometric Functions of Any Angle . . . . . . . . . . . . . . . . . . . . . . . 24

1.5

Rotations and Reflections of Angles . . . . . . . . . . . . . . . . . . . . . . . . . 32

2

General Triangles

38

2.1

The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2

The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3

The Law of Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.4

The Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5

Circumscribed and Inscribed Circles . . . . . . . . . . . . . . . . . . . . . . . . 59

3

Identities

65

3.1

Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2

Sum and Difference Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3

Double-Angle and Half-Angle Formulas . . . . . . . . . . . . . . . . . . . . . . 78

3.4

Other Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4

Radian Measure

87

4.1

Radians and Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2

Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3

Area of a Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4

Circular Motion: Linear and Angular Speed . . . . . . . . . . . . . . . . . . . . 100

5

Graphing and Inverse Functions

103

5.1

Graphing the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 103

5.2

Properties of Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . 109

5.3

Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6

Additional Topics

129

6.1

Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2

Numerical Methods in Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . 133

v

vi

CONTENTS

6.3

Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.4

Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Appendix A:

Answers and Hints to Selected Exercises

152

Appendix B:

Graphing with Gnuplot

155

GNU Free Documentation License

160

History

168

Index

169

1 Right Triangle Trigonometry

Trigonometry is the study of the relations between the sides and angles of triangles. The

word “trigonometry” is derived from the Greek words trigono (τρ´ιγωνo), meaning “triangle”,

and metro (µǫτρ ´

ω), meaning “measure”. Though the ancient Greeks, such as Hipparchus

and Ptolemy, used trigonometry in their study of astronomy between roughly 150 B.C. - A.D.

200, its history is much older. For example, the Egyptian scribe Ahmes recorded some rudi-

mentary trigonometric calculations (concerning ratios of sides of pyramids) in the famous

Rhind Papyrus sometime around 1650 B.C.1

Trigonometry is distinguished from elementary geometry in part by its extensive use of

certain functions of angles, known as the trigonometric functions. Before discussing those

functions, we will review some basic terminology about angles.

1.1 Angles

Recall the following definitions from elementary geometry:

(a) An angle is acute if it is between 0◦ and 90◦.

(b) An angle is a right angle if it equals 90◦.

(c) An angle is obtuse if it is between 90◦ and 180◦.

(d) An angle is a straight angle if it equals 180◦.

(a) acute angle

(b) right angle

(c) obtuse angle

(d) straight angle

Figure 1.1.1

Types of angles

In elementary geometry, angles are always considered to be positive and not larger than

360◦. For now we will only consider such angles.2 The following definitions will be used

throughout the text:

1Ahmes claimed that he copied the papyrus from a work that may date as far back as 3000 B.C.

2Later in the text we will discuss negative angles and angles larger than 360◦.

1

2

Chapter 1 • Right Triangle Trigonometry

§1.1

(a) Two acute angles are complementary if their sum equals 90◦. In other words, if 0◦ ≤

A , ∠ B ≤ 90◦ then ∠ A and ∠ B are complementary if ∠ A + ∠ B = 90◦.

(b) Two angles between 0◦ and 180◦ are supplementary if their sum equals 180◦. In other

words, if 0◦ ≤ ∠ A , ∠ B ≤ 180◦ then ∠ A and ∠ B are supplementary if ∠ A + ∠ B = 180◦.

(c) Two angles between 0◦ and 360◦ are conjugate (or explementary) if their sum equals

360◦. In other words, if 0◦ ≤ ∠ A , ∠ B ≤ 360◦ then ∠ A and ∠ B are conjugate if ∠ A+∠ B =

360◦.

B

B

A

A

A

B

(a) complementary

(b) supplementary

(c) conjugate

Figure 1.1.2

Types of pairs of angles

Instead of using the angle notation ∠ A to denote an angle, we will sometimes use just a

capital letter by itself (e.g. A, B, C) or a lowercase variable name (e.g. x, y, t). It is also common to use letters (either uppercase or lowercase) from the Greek alphabet, shown in

the table below, to represent angles:

Table 1.1

The Greek alphabet

Letters

Name

Letters

Name

Letters

Name

A

α

alpha

I

ι

iota

P

ρ

rho

B

β

beta

K

κ

kappa

Σ

σ

sigma

Γ

γ

gamma

Λ

λ

lambda

T

τ

tau

δ

delta

M

µ

mu

Υ

υ

upsilon

E

ǫ

epsilon

N

ν

nu

Φ

φ

phi

Z

ζ

zeta

Ξ

ξ

xi

X

χ

chi

H

η

eta

O

o

omicron

Ψ

ψ

psi

Θ

θ

theta

Π

π

pi

ω

omega

In elementary geometry you learned that the sum of the angles in a triangle equals 180◦,

and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a

right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles

is 90◦ and the other two angles are acute angles whose sum is 90◦ (i.e. the other two angles

are complementary angles).

Angles • Section 1.1

3

Example 1.1

For each triangle below, determine the unknown angle(s):

E

Y

B

53◦

3 α

35◦

20◦

α

α

A

C

D

F

X

Z

Note: We will sometimes refer to the angles of a triangle by their vertex points. For example, in the

first triangle above we will simply refer to the angle ∠ BAC as angle A.

Solution: For triangle △ ABC, A = 35◦ and C = 20◦, and we know that A + B + C = 180◦, so 35◦ + B + 20◦ = 180◦

B = 180◦ − 35◦ − 20◦

B = 125◦ .

For the right triangle △ DEF, E = 53◦ and F = 90◦, and we know that the two acute angles D and E

are complementary, so

D + E = 90◦

D = 90◦ − 53◦

D = 37◦ .

For triangle △ X Y Z, the angles are in terms of an unknown number α, but we do know that X + Y +

Z = 180◦, which we can use to solve for α and then use that to solve for X , Y , and Z:

α + 3 α + α = 180◦

5 α = 180◦

α = 36◦

X = 36◦ , Y = 3 × 36◦ = 108◦ , Z = 36◦

Example 1.2

Thales’ Theorem states that if A, B, and C are (distinct) points on a circle such that the line segment

AB is a diameter of the circle, then the angle ∠ ACB is a right angle (see Figure 1.1.3(a)). In other words, the triangle △ ABC is a right triangle.

C

C

α β

α

β

A

B

A

B

O

O

(a)

(b)

Figure 1.1.3

Thales’ Theorem: ∠ ACB = 90◦

To prove this, let O be the center of the circle and draw the line segment OC, as in Figure 1.1.3(b).

Let α = ∠ BAC and β = ∠ ABC. Since AB is a diameter of the circle, OA and OC have the same length (namely, the circle’s radius). This means that △ OAC is an isosceles triangle, and so ∠ OC A =

OAC = α. Likewise, △ OBC is an isosceles triangle and ∠ OCB = ∠ OBC = β. So we see that

ACB = α+ β. And since the angles of △ ABC must add up to 180◦, we see that 180◦ = α+( α+ β)+ β =

2 ( α + β), so α + β = 90◦. Thus, ∠ ACB = 90◦.

QED

4

Chapter 1 • Right Triangle Trigonometry

§1.1

In a right triangle, the side opposite the right angle is called the hy-

B

potenuse, and the other two sides are called its legs. For example, in

c

Figure 1.1.4 the right angle is C, the hypotenuse is the line segment

a

AB, which has length c, and BC and AC are the legs, with lengths a

and b, respectively. The hypotenuse is always the longest side of a right

A

b

C

triangle (see Exercise 11).

Figure 1.1.4

By knowing the lengths of two sides of a right triangle, the length of

the third side can be determined by using the Pythagorean Theorem:

Theorem 1.1. Pythagorean Theorem: The square of the length of the hypotenuse of a

right triangle is equal to the sum of the squares of the lengths of its legs.

Thus, if a right triangle has a hypotenuse of length c and legs of lengths a and b, as in

Figure 1.1.4, then the Pythagorean Theorem says:

a 2 + b 2 = c 2

(1.1)

Let us prove this. In the right triangle △ ABC in Figure 1.1.5(a) below, if we draw a line

segment from the vertex C to the point D on the hypotenuse such that CD is perpendicular

to AB (that is, CD forms a right angle with AB), then this divides △ ABC into two smaller

triangles △ CBD and △ ACD, which are both similar to △ ABC.

B

c

d

C

D

B

a

d

b

a

c

d

A

b

C

C

D

A

c d

D

(a) ABC

(b) CBD

(c) ACD

Figure 1.1.5

Similar triangles △ ABC, △ CBD, △ ACD

Recall that triangles are similar if their corresponding angles are equal, and that similarity

implies that corresponding sides are proportional. Thus, since △ ABC is similar to △ CBD,

by proportionality of corresponding sides we see that

c

a

AB is to CB (hypotenuses) as BC is to BD (vertical legs)

=

cd = a 2 .

a

d

Since △ ABC is similar to △ ACD, comparing horizontal legs and hypotenuses gives

b

c

=

b 2 = c 2 − cd = c 2 − a 2

a 2 + b 2 = c 2 . QED

c d

b

Note: The symbols ⊥ and ∼ denote perpendicularity and similarity, respectively. For exam-

ple, in the above proof we had CD AB and △ ABC ∼ △ CBD ∼ △ ACD.

Angles • Section 1.1

5

Example 1.3

For each right triangle below, determine the length of the unknown side:

B

Y

E

5

z

a

2

1

1

A

4

C

D

e

F

X

1

Z

Solution: For triangle △ ABC, the Pythagorean Theorem says that

a 2 + 42 = 52

a 2 = 25 − 16 = 9

a = 3 .

For triangle △ DEF, the Pythagorean Theorem says that

e 2 + 12 = 22

e 2 = 4 − 1 = 3

e =

3 .

For triangle △ X Y Z, the Pythagorean Theorem says that

12 + 12 = z 2

z 2 = 2

z =

2 .

Example 1.4

A 17 ft ladder leaning against a wall has its foot 8 ft from the base of the wall. At

what height is the top of the ladder touching the wall?

Solution: Let h be the height at which the ladder touches the wall. We can as-

17

sume that the ground makes a right angle with the wall, as in the picture on the

h

right. Then we see that the ladder, ground, and wall form a right triangle with a

hypotenuse of length 17 ft (the length of the ladder) and legs with lengths 8 ft and

90◦

h ft. So by the Pythagorean Theorem, we have

8

h 2 + 82 = 172

h 2 = 289 − 64 = 225

h = 15 ft .

Exercises

For Exercises 1-4, find the numeric value of the indicated angle(s) for the triangle △ ABC.

1. Find B if A = 15◦ and C = 50◦.

2. Find C if A = 110◦ and B = 31◦.

3. Find A and B if C = 24◦, A = α, and B = 2 α.

4. Find A, B, and C if A = β and B = C = 4 β.

For Exercises 5-8, find the numeric value of the indicated angle(s) for the right triangle △ ABC, with

C being the right angle.

5. Find B if A = 45◦.

6. Find A and B if A = α and B = 2 α.

7. Find A and B if A = φ and B = φ 2.

8. Find A and B if A = θ and B = 1/ θ.

9. A car goes 24 miles due north then 7 miles due east. What is the straight distance between the

car’s starting point and end point?

6

Chapter 1 • Right Triangle Trigonometry

§1.1

10. One end of a rope is attached to the top of a pole 100 ft high. If the rope is 150 ft long, what is

the maximum distance along the ground from the base of the pole to where the other end can be

attached? You may assume that the pole is perpendicular to the ground.

11. Prove that the hypotenuse is the longest side in every right triangle. ( Hint: Is a 2 + b 2 > a 2 ? ) 12. Can a right triangle have sides with lengths 2, 5, and 6? Explain your answer.

13. If the lengths a, b, and c of the sides of a right triangle are positive integers, with a 2 + b 2 = c 2, then they form what is called a Pythagorean triple. The triple is normally written as ( a, b, c).

For example, (3,4,5) and (5,12,13) are well-known Pythagorean triples.

(a) Show that (6,8,10) is a Pythagorean triple.

(b) Show that if ( a, b, c) is a Pythagorean triple then so is ( ka, kb, kc) for any integer k > 0. How would you interpret this geometrically?

(c) Show that (2 mn, m 2 − n 2, m 2 + n 2) is a Pythagorean triple for all integers m > n > 0.

(d) The triple in part(c) is known as Euclid’s formula for generating Pythagorean triples. Write

down the first ten Pythagorean triples generated by this formula, i.e. use: m = 2 and n = 1;

m = 3 and n = 1, 2; m = 4 and n = 1, 2, 3; m = 5 and n = 1, 2, 3, 4.

14. This exercise will describe how to draw a line through any point outside a circle such that the

line intersects the circle at only one point. This is called a tangent line to the circle (see the picture

on the left in Figure 1.1.6), a notion which we will use throughout the text.

tangent line

A

O

P

C

not tangent

Figure 1.1.6

On a sheet of paper draw a circle of radius 1 inch, and call the center of that circle O. Pick a

point P which is 2.5 inches away from O. Draw the circle which has OP as a diameter, as in the

picture on the right in Figure 1.1.6. Let A be one of the points where this circle intersects the first circle. Draw the line through P and A. In general the tangent line through a point on a circle

is perpendicular to the line joining that point to the center of the circle (why?). Use this fact to

explain why the line you drew is the tangent line through A and to calculate the length of P A.

Does it match the physical measurement of P A?

15. Suppose that

C

ABC is a triangle with side AB a diameter of a circle

with center O, as in the picture on the right, and suppose that the

vertex C lies on the circle. Now imagine that you rotate the circle 180◦

around its center, so that △ ABC is in a new position, as indicated by

A

B

the dashed lines in the picture. Explain how this picture proves Thales’

O

Theorem.

Trigonometric Functions of an Acute Angle • Section 1.2

7

1.2 Trigonometric Functions of an Acute Angle

Consider a right triangle △ ABC, with the right angle at C and

e

B o

with lengths a, b, and c, as in the figure on the right. For the acute

pp

potenus

c

o

angle A, call the leg BC its opposite side, and call the leg AC its

hy

a sit

adjacent side. Recall that the hypotenuse of the triangle is the side

e

AB. The ratios of sides of a right triangle occur often enough in prac-

A

b

C

tical applications to warrant their own names, so we define the six

adjacent

trigonometric functions of A as follows:

Table 1.2

The six trigonometric functions of A

Name of function

Abbreviation

Definition

opposite side

a

sine A

sin A

=

=

hypotenuse

c

adjacent side

b

cosine A

cos A

=

=

hypotenuse

c

opposite side

a

tangent A

tan A

=

=

adjacent side

b

hypotenuse

c

cosecant A

csc A

=

=

opposite side

a

hypotenuse

c

secant A

sec A

=

=

adjacent side

b

adjacent side

b

cotangent A

cot A

=

=

opposite side

a

We will usually use the abbreviated names of the functions. Notice from Table 1.2 that

the pairs sin A and csc A, cos A and sec A, and tan A and cot A are reciprocals:

1

1

1

csc A =

sec A =

cot A =

sin A

cos A

tan A

1

1

1

sin A =

cos A =

tan A =

csc A

sec A

cot A

8

Chapter 1 • Right Triangle Trigonometry

§1.2

Example 1.5

For the right triangle △ ABC shown on the right, find the values of all six trigono-

B

metric functions of the acute angles A and B.

5

3

Solution: The hypotenuse of △ ABC has length 5. For angle A, the opposite side

BC has length 3 and the adjacent side AC has length 4. Thus:

A

4

C

opposite

3

adjacent

4

opposite

3

sin A =

=

cos A =

=

tan A =

=

hypotenuse

5

hypotenuse

5

adjacent

4

hypotenuse

5

hypotenuse

5

adjacent

4

csc A =

=

sec A =

=

cot A =

=

opposite

3

adjacent

4

opposite

3

For angle B, the opposite side AC has length 4 and the adjacent side BC has length 3. Thus:

opposite

4

adjacent

3

opposite

4

sin B =

=

cos B =

=

tan B =

=

hypotenuse

5

hypotenuse

5

adjacent

3

hypotenuse

5

hypotenuse

5

adjacent

3

csc B =

=

sec B =

=

cot B =

=

opposite

4

adjacent

3

opposite

4

Notice in Example 1.5 that we did not specify the units for the lengths. This raises the

possibility that our answers depended on a triangle of a specific physical size.

For example, suppose that two different students are reading this textbook: one in the

United States and one in Germany. The American student thinks that the lengths 3, 4, and

5 in Example 1.5 are measured in inches, while the German student thinks that they are

measured in centimeters. Since 1 in ≈ 2.54 cm, the students are using triangles of different

physical sizes (see Figure 1.2.1 below, not drawn to scale).