OVERVIEW:
This section of Chapter 4 discusses the geometrical
origin of the "periodic" property of the six trig functions.
Saying that a function like sin(θ) or sec(θ)
is "periodic in θ" means that, as one steadily increases the angle
θ, the function sin(θ) or sec(θ)
repeats itself over a regular interval.
- The smallest increase, δθ, in θ
such that sin(θ + δθ)=sin(θ) FOR
ALL θ is called the period of the sine function
- Similarly, the smallest increase, δθ, in θ
such that sec(θ + δθ)=sec(θ) FOR
ALL θ is called the period of the secant function,
(with analogous definitions for the periods of the other
four trig functions)
- Note that, once you know that the function of interest is
periodic, you only have to figure out what its graph looks like
over a region of θ covering one period (one full repetition).
This part of the graph then just gets repeated endlessly for other θ.
A list of the background material needed to follow
the contents of this section is given below. If you are unfamiliar
with one of the background topics, or feel you would like
to review it, click on the relevant link.
If you are already familar with all of the background material,
you can proceed directly to the contents of the current section
of the course by clicking
HERE.
BACKGROUND MATERIAL (AND LINKS):
For this section of the course, you should be familiar with
- the idea and geometrical meaning of angles and angular measure (reviewable
by reading (or re-reading) the various sections of Chapter 1 of the course,
which can be accessed by clicking
HERE),
- the basic geometrical definitions of all six trig functions
as the ratios of sides of the appropriate right angle triangles,
phrased in terms of the concepts "hypoteneuse", "adjacent" and
"opposite" (reviewable by reading (or re-reading)
Section (c) of Chapter 2
of the course)
- the algebraic relations amongst the various trig functions
of the same angle, especially those which allow the tangent, cotangent,
secant and cosecant to be written in terms of the sine and cosine
(reviewable by reading (or re-reading)
Section (a) of Chapter 3
of this course) and
- the unit circle picture for the sine and cosine functions
(reviewable by reading (or re-reading)
Section (b) of Chapter 3
of this course)
- the generalization of the trig functions to
angles corresponding to non-first-quadrant directions
using the unit circle picture for the sine and cosine
functions and the algebraic expressions for the tangent, cotangent,
secant and cosecant functions as ratios involving the sine and cosine
(reviewable by reading (or re-reading)
Section
(c) of Chapter 3
of this course)