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
iii
1
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Trigonometric Functions of an Acute Angle . . . . . . . . . . . . . . . . . . . .
7
Applications and Solving Right Triangles . . . . . . . . . . . . . . . . . . . . . 14
Trigonometric Functions of Any Angle . . . . . . . . . . . . . . . . . . . . . . . 24
Rotations and Reflections of Angles . . . . . . . . . . . . . . . . . . . . . . . . . 32
38
The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
The Law of Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
The Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Circumscribed and Inscribed Circles . . . . . . . . . . . . . . . . . . . . . . . . 59
65
Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Sum and Difference Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Double-Angle and Half-Angle Formulas . . . . . . . . . . . . . . . . . . . . . . 78
Other Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
87
Radians and Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Area of a Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Circular Motion: Linear and Angular Speed . . . . . . . . . . . . . . . . . . . . 100
Graphing and Inverse Functions
103
Graphing the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 103
Properties of Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . 109
Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 120
129
Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Numerical Methods in Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . 133
v
vi
CONTENTS
Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Answers and Hints to Selected Exercises
152
155
GNU Free Documentation License
160
168
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
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 hy