KIM MALTMAN'S MINI-COURSE ON TRIG BASICS
This mini-course on the basics of trigonometry aims to provide
students with an easy-to-grasp, intuitive understanding of
- the meaning of angles and angular measure (a topic/concept crucial
to the rest of the course);
- the fundamental geometrical meaning of the six trigonometric (trig)
functions (the sine, cosine, secant, cosecant, tangent and cotangent);
- the key algebraic relations amongst these functions; and
- a number of important, and commonly used, properties of these
functions based on their geometrical meanings and algebraic interrelations,
including
(i) how to use elementary geometry, in combination with the basic
geometrical definitions of the trig functions, to work out exact values
for these functions for certain special angles;
(ii) how each of the six trig functions changes as the angle θ on which
it depends changes; (iii) the geometrical origins of a number of important
"trig identities", and how one goes about actually using and understanding
the meaning of these identities.
A basic outline of the course is given in the Table of Contents.
For more on how the material in the course proper
is organized, see the comments following the Table of Contents.
TABLE OF CONTENTS
Chapter 1:
The Basics of Angles and Angular Measure
- (a)
Motivation for the use of angles and angular measure to specify
directions in the plane
- (b)
Basic conventions for assigning angles (numbers) to directions
in the plane
- (c)
Degrees and radians as units of angular measure, and how to
convert from one to the other
- (d)
Why there is more than one angle corresponding to a given direction
in the plane
Chapter 2:
Key Geometrical Background and the Fundamental Geometrical Meaning of
the Six Trig Functions
- (a)
Demonstration of the fact that the sum of the interior angles of any triangle
is equal to π radians (180o)
- (b)
Pythagoras' Theorem (including a simple geometric proof)
- (c)
The basic geometric definitions of the six trigonometric functions
as ratios of sides of right-angle triangles
- (d)
Exact values of the trig functions of the angles π/6, π/4 and π/3
radians from elementary geometry
Chapter 3:
Generalizing the Trig Functions to Non-First-Quadrant Directions
- (a)
Algebraic interrelations amongst the six trig functions
- (b)
The unit circle picture of the geometry of the sine and cosine
functions
- (c)
Using the unit circle picture to generalize the definitions
of the trig functions to angles outside of those first quadrant
angles which naturally occur as the interior angles of right
angle triangles
- (d)
An example of the utility of the generalized unit circle definitions:
trig functions values for directions along the +x, -x,
+y and -y axes
- (e)
An alternate triangle-based perspective for the generalization of
the trig functions to non-first quadrant angles
- (f)
Exact values from elementary geometry
for the trig functions of angles corresponding to
directions one-third, one-half or two-thirds of the way through
the second, third and fourth quadrants
Chapter 4:
Range, Periodicity, and How the Trig Functions Change With Angle
- (a)
The behavior of the six trig functions for angles corresponding
to directions in between the special directions for which exact values
were already worked out above using elementary geometry
- (b)
The "ranges" (sets of all possible allowed values)
of the six trig functions
- (c)
The "periodic" (regular repeating) properties of the six trig functions
Chapter 5:
Trig Identities and Their Implications
- (a)
Background discussion on the idea of an "identity", and how it differs
from the more general idea of an "equation"
- (b)
Three important identities which follow, via elementary geometry,
from Pythagoras' Theorem
- (c)
A preliminary summary of the identities to be discussed in Sections
(d), (e) and (f) of this chapter, including some basic discussion and
illustrations of how these identities are meant to be interpreted
and used
- (d)
Identities between trig functions of pairs of angles associated
with different, but closely related, right-angle triangles
- (e)
The sine and cosine additional formulas
- (f)
Other identities easily worked out from the sine and cosine addition
formulas, e.g., the tangent and cotangent addition formulas, the double-angle
formulas, the half-angle formulas, ...
ORGANIZATION OF THE MATERIAL IN THE COURSE
The course consists of five chapters, each containing a number
of sections. Each such section focusses on a key topic or small set of
related topics. The material is organized as follows:
- The links above for each chapter take you to a "chapter summary page".
The chapter summary pages contain:
- a short overview of what is to be covered in the chapter,
- links to earlier sections of the course which are of particular
relevance as background to the material in the current chapter,
- links to take you to the materials in the various sections
of the chapter in question, and
- a link to the PDF file containing answers to the exercises
from all sections of the current chapter. (The set of
answers to the exercises in a given chapter form the last section of
that chapter.)
- In most cases, the link to a particular section of a given chapter
(found both in the Table of Contents above, and on the summary page for the
chapter as a whole) will take you to another summary page
specific to that section of the chapter. This "chapter section summary page"
will contain:
- a short overview of what is to be covered in the section,
- links to earlier sections of the course which are of particular
relevance as background to the material in the current section,
- a link to the PDF file containing the actual material
in the section, and
- another link to the PDF file containing the answers to the exercises
from all sections of the current chapter.
- There are two sections (Section A of Chapter 1 and Section A of Chapter
5) which provide only background motivation, without introducing any new
material. These sections have no "chapter section summary page". The
associated links in the Table of Contents and on the relevant
"chapter summary page" take you directly to the PDF files containing
the material in question in these cases.
- At the end of each section of the course (apart from the special
"motivational" sections, Section A of Chapter 1 and Section A of Chapter 5)
are exercises which students reading the section are meant to
try for themselves. The answers to the exercises for all sections
of the chapter in question are given in the last section of the chapter.
Links to the PDF file containing these answers can be found at
the bottom of the relevant "chapter section summary page" as well
as at the bottom of the "chapter summary page".
