# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 3, SECTION E: AN ALTERNATIVE WAY (INVOLVING RIGHT-ANGLE TRIANGLES) OF THINKING ABOUT THE GENERALIZATION OF THE TRIG FUNCTIONS TO NON-FIRST-QUADRANT ANGLES

OVERVIEW:

This section of Chapter 3 discusses an alternate geometrical picture for thinking about how to generalize the trig functions from first quadrant angles to angles corresponding to directions in all other quadrants.

• This new picture involves taking a familiar right-angle triangle whose hypoteneuse points in a first quadrant direction (and hence corresponds naturally to a first quadrant angle) and appropriately "flopping it over" to create a new triangle whose hypotenteuse points in a second, third or fourth quadrant direction.

• We then show how, by changing the signs of the "lengths" of one (or both) of the non-hypoteneuse sides of the original triangle and assigning them as "lengths" of the corresponding sides of the new second, third or fourth quadrant triangle, the generalized trig function values for the new second, third and fourth quadrant directions can be obtained using the same geometrial "ratio of sides" definitions as were used for the first quadrant angles

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.

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 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)