# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 3, SECTION F: EXACT VALUES FOR THE TRIG FUNCTIONS OF ANGLES 1/3, 1/2 AND 2/3 OF THE WAY THROUGH QUADRANTS II, III AND IV

OVERVIEW:

This section of Chapter 3 shows how to use

• the alternate geometrical ("flopped triangle") picture for thinking about the trig functions of non-first-quadrant angles; and

• the exact values previously worked out for the trig functions of the first quadrant angles π/6, π/4 and π/3 radians

to also work out the exact values of all six trig functions of angles one-third, one-half and two-thirds of the way through Quadrants II, III and IV. These angles are special in the sense of corresponding to right-angle triangles related to first quadrant right-angle triangles with an interior angle equal to either π/6, π/4 or π/3 (radians) by the "flopping-over" construction discussed in detail in the previous section, Section (e), of this chapter.

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):

To follow the material in 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)

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

• the values of the trig functions for the special first quadrant angles π/6, π/4 and π/3 (or, better yet, the elementary geometrical constructions that produce the sample π/4 and π/6,π/3 triangles from which these values are generated) (reviewable by reading (or re-reading) Section (d) of Chapter 2 of this course)

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