# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 4, SECTION C: THE PERIODICITY (REPEATING PROPERTY) OF THE SIX TRIG FUNCTIONS

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)

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