# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 4, SECTION A: HOW DO THE TRIG FUNCTIONS CHANGE IN BETWEEN THE 16 SPECIAL DIRECTIONS FOR WHICH WE KNOW THEIR EXACT VALUES?

OVERVIEW:

This section of Chapter 4 deals with the general question of how the trig functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ) and cot(θ) change as the angle θ is changed.

This information will allow you to easily figure out how to generate (reproduce) the graphs of the six trig functions as functions of θ which most of you will have seen in your high school textbooks. Past experience suggests that a significant number of you, even if you have been shown these graphs, will not have had explained to you where it is that they come from. After working your way through this section of the course, you will be able to easily generate such graphs for yourself, and will no longer have to try to memorize them.

The key pieces of information which will be used in figuring out how the trig functions change as θ is changed are:

• the unit circle picture for the sine and cosine functions (in its general form, where it applies to ALL directions in the plane); and

• the algebraic relations which express the tangent, cotangent, secant and cosecant as ratios involving the sine and cosine, and which define the generalizations of these functions to non-first-quadrant directions.

We will find out, in the end, that

• Each of the trig functions behaves very simply in each of the four quadrants, either steadily increasing or steadily decreasing as one increases θ through the quadrant in question.

• Whether a particular trig function increases or decreases as one increases θ through a particular quadrant depends on which function it is we are considering, and which quadrant it is we are focussing on.

• It is easy to use the basic geometry provided by the unit circle picture to find out whether the sine or cosine increases or decreases in a given quadrant.

• The information about how the sine and cosine behave (increasing or decreasing) in the quadrant of interest, together with the information about how the tangent, cotangent, secant and cosecant are expressed in terms of the sine and/or cosine, make it easy to find out how the tangent, cotangent, secant or cosecant behave (increasing or decreasing) in the same quadrant.

Note that we already know the values of all six trig functions for three angles in each quadrant (as well as along the four coordinate directions), so, once we know that each of the functions either steadily increases or steadily decreases as we move through each of the quadrants, we can also use our knowledge of the exact values in the quadrant of interest to help remind ourselves of whether the function of interest is decreasing or increasing in that quadrant.

It is, however, a good idea to go through the discussion of the underlying geometry once, as done in this section of the course, so you acquire a good intuitive understanding of the geometrical origin of this behavior of the trig functions, and can reproduce the results for yourself in future if you need to.

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 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 the course);

• the unit circle picture for the sine and cosine functions (reviewable by reading (or re-reading) Section (b) of Chapter 3 of the 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 the course)

Knowledge (or a brief review) of the geometrical constructions and special triangles which allow you to reconstitute the values of all six trig functions for the sixteen special directions around the plane for which these values are known exactly will also be helpful. This material can be reviewed by reading (or re-reading) Section (d) of Chapter 2, Section (e) of Chapter 3 and Section (f) of Chapter 3 of the course.

CLICK HERE TO PROCEED TO THE CURRENT SECTION OF THE COURSE
CLICK HERE FOR TO SEE THE ANSWERS TO THE EXERCISES FOR ALL SECTIONS OF CHAPTER 4 INCLUDING THIS SECTION