# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 5, SECTION D: IDENTITIES BETWEEN TRIG FUNCTIONS OF PAIRS OF ANGLES ASSOCIATED WITH DIFFERENT, BUT CLOSELY RELATED, RIGHT-ANGLE TRIANGLES

OVERVIEW:

This section of Chapter 5 deals with identities of the type where "old" information (in this case, the values of the trig functions of an "original" angle) is used to simplify the determination of analogous "new" information (in this case, the values of the trig functions of a set of "new" angles which have one of four special geometrical relations to the old angle).

The specific topics to be covered are:

• Relations involving the two different non-right angles, θ and (π/2)-θ, lying in the same right-angle triangle (the "old" angle here being θ and the "new" angle (π/2)-θ)

• Relations involving the "old" angle θ and the "new" angle π-θ (when θ is a first quadrant angle π-θ is just the related second quadrant angle associated with the right-angle triangle obtained by "flopping-over" the first quadrant θ triangle into the second quadrant, as discussed in Section (e) of Chapter 3 of the course)

• Relations involving the "old" angle θ and the "new" angle -θ (when θ is a first quadrant angle, -θ is just the related fourth quadrant angle associated with the right-angle triangle obtained by "flopping-over" the first quadrant θ triangle into the fourth quadrant, as discussed in Section (e) of Chapter 3 of the course)

• Relations involving the "old" angle θ and the "new" angle π+θ (when θ is a first quadrant angle π+θ is just the related third quadrant angle associated with the right-angle triangle obtained by "double-flopping" the first quadrant θ triangle into the third quadrant, as discussed in Section (e) of Chapter 3 of the course)

If you have not already done so, it would be a good idea, before continuing here, to take a look at Section (c) of this chapter of the course, where some concrete illustrations of why it is that this type of identity can be useful are given.

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 quickly review it, click on the relevant link.

If you are already familiar with this background, 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 with

• the fact that the sum of the interior angles of any triangle is π radians and hence that, for a right angle triangle having one non-right angle θ, the other non-right angle is necessarily equal to (π/2)-θ (reviewable by reading (or re-reading) Section (a) of Chapter 2 of the course);

• the basic geometrical definitions of all six trig functions as the ratios of sides of 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 inter-relations amongst the six trig functions which allow the tangent, cotangent, secant and cosecant functions to be written as ratios involving the sine and cosine (reviewable by reading (or re-reading) Section (a) of Chapter 3 of the course);

• the unit circle picture for generalizing the sine and cosine functions, and from these, also the tangent, cotangent, secant and cosecant functions, from Quadrant I to Quadrant II, III and IV values of the relevant angle (reviewable by reading (or re-reading) Section (b) and Section (c) of Chapter 3 of the course); and

• the alternate "flopped-over" triangle perspective for thinking about the relation between trig functions of first quadrant angles and the appropriately related second, third and four quadrant angles (reviewable by reading (or re-reading) Section (e) of Chapter 3 of the course).

Other more basic background topics which it may be helpful to review if you are coming to any of the above background material sections for the first time are

• the idea and geometrical meaning of angles and angular measure (reviewable by reading (or re-reading) the topics accessible via the links on the main page of Chapter 1 of the course); and

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