# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 3:GENERALIZING THE SIX BASIC TRIG FUNCTIONS TO NON-FIRST QUADRANT DIRECTIONS

OVERVIEW:

The main topic of this chapter is how one goes about naturally generalizing the idea of the trig functions from the case of angles corresponding to first quadrant directions (angles which natural occur as the interior angles of right angle triangles) to the case of angles corresponds to a non-first-quadrant directions. Topics to be covered are:

• Algebraic relations between the different trig functions of the same angle

• The "unit circle picture" for the sine and cosine (an alternate way of visualizing the basic geometric meaning of the sine and cosine functions)

• Using the generalization of the sine and cosine functions, together with the algebraic relations giving the other four trig functions in terms of the sine and cosine to also generalize the tangent, cosecant, secant and cotangent functions to non-first-quadrant directions

• An example of the utility of the unit circle perspective: the sine and cosine values of directions along the +x, -x, +y and -y axis directions

• An alternate way of thinking about the generalization to Quadrant II, III, and IV directions involving taking Quadrant I triangles and "flopping them over" into Quadrant II, III or IV

• Examples of the generalization to other quadrants: exact values of the trig functions for angles corresponding to directions one-third, one-half and two-thirds of the way through Quadrants II, III, and IV from those for the corresponding Quadrant I angles π/6, π/4 and π/3

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 3 TOPICS" sections below.

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

• the idea and geometrical meaning of angles and angular measure (this material can be reviewed, if needed, by reading (or re-reading) Chapter 1 of the course), and

• the basic geometrical definitions of all six trig functions as the ratios of sides of the appropriate right angle triangles (this material can be reviewed, if needed, by reading (or re-reading) Section (c) of Chapter 2 of the course)

It will also be useful to remember (or briefly review) the construction of examples of those "special" right angle triangles having an internal angle equal to π/6, π/4 or π/3, from which one can get exact values for all six trig functions of any of these three special angles (this material can be reviewed by taking a look at Section (d) of Chapter 2 of this course)

LINKS FOR THE NEW CHAPTER 3 TOPICS:

Click on the link for the appropriate topic below for more information on that topic. In general, it would be a good idea to go through the material in the order listed, unless you are returning to this page for a refresher on a particular topic (or topics) or are already familiar with some of the topics from previous review.