# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 4:SOME GENERAL PROPERTIES OF THE TRIG FUNCTIONS

OVERVIEW:

This chapter deals with some general features of the trig functions which emerge when one considers how they change as the angle on which they depend is changed. Topics to be covered are:

• What happens to the various trig functions for angles corresponding to directions in between those for which we know the exact values?

• The ranges (complete set of possible values) of the various trig functions

• The periodicity (repeating property) of the trig functions

Links to files dealing with each of these new topics, as well as links to earlier sections of the course which are particularly relevant as background, are given in the "BACKGROUND MATERIAL LINKS" and "LINKS FOR THE NEW CHAPTER 4 TOPICS" sections below.

The key background needed for this Chapter of the course is a familiarity with

• the algebraic relations expressing the tangent, cotangent, secant and cosecant functions in terms of the sine and cosine (reviewable by reading (or re-reading) Section (a) of Chapter 3 of the course)

• the unit circle picture for generalizing the sine and cosine functions, and from these, also the tangent, cotangent, secant and cosecant functions, from Quadrant I to Quadrant II, III and IV values of the relevant angle (reviewable by reading (or re-reading) Section (b) and Section (c) of Chapter 3 of the course)

It will also be useful to be familiar with (or briefly review) what we have already learned about obtaining exact values for the trig functions of the sixteen previously studied special directions in the plane, namely

• the construction of examples of those "special" right angle triangles having an internal angle equal to π/6, π/4 or π/3 and the process of using these to obtain exact values for all six trig functions of these angles (reviewable by reading (or re-reading) Section (d) of Chapter 2 of the course);

• the process of using the results for the trig functions of π/6, π/4 or π/3 to obtain exact values for six trig functions of the related angles one-third, one-half, and two-thirds of the way through Quadrants II, III and IV (reviewable by reading (or re-reading) Section (e) and, more directly, Section (f) of Chapter 3 of the course); and

• the process of using of the unit circle picture to work out the trig functions of angles corresponding to directions along the +x, -x, +y or -y axes (reviewable by reading (or re-reading) Section (d) of Chapter 3 of the course).

Additional more basic background topics, which you should also be familiar with, are:

• the basic idea and geometrical meaning of angles and angular measure (reviewable by reading (or re-reading) the relevant topics from Chapter 1 of the course, which are accessible by clicking HERE), and

• the basic geometrical definitions of the six trig functions in terms of the concepts "hypoteneuse", "opposite" and "adjacent" (reviewable by reading (or re-reading) Section (c) of Chapter 2 of the course)

LINKS FOR THE NEW CHAPTER 4 TOPICS