# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 4, SECTION B: THE RANGES OF THE SIX TRIG FUNCTIONS

OVERVIEW:

This section of Chapter 4 discusses the set of all possible values that are allowed for each of the six trig functions when one varies the angle θ on which they depend over values corresponding to all possible physical directions. We will find that

• The possible values of tan(θ) and cot(θ) are unrestricted (i.e., given any number κ it is always possible to find angles α for which tan(α)=κ and angles β for which cot(β)=κ).

• The possible values of sin(θ) and cos(θ) are restricted to lie between -1 and +1 (i.e., -1 ≤ sin(θ) ≤ +1 and, -1 ≤ cos(θ) ≤ +1 for all angles θ).

• The possible values of sec(θ) and csc(θ) are restricted to be either ≥ +1 or ≤ -1 for all angles θ.

We will also discuss what type of directions correspond to angles giving each of the possible allowed values of each of the six trig functions.

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 the 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 the course)

• the alternate "flopped triangle" approach to generalizing the trig functions from first quadrant angles to angles corresponding to second, third and fourth quadrant directions (reviewable by reading (or re-reading) Section (e) and Section (f) of Chapter 3 of the course)

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