# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 5, SECTION E: THE SINE AND COSINE ADDITIONAL FORMULAS

OVERVIEW:

This section of Chapter 5 discusses and proves a set of important trig identities called the sine and cosine addition formulas. These formulas show you how to work out the values of the sine and cosine of two new angles, θ1 + θ2 and θ1 - θ2, provided you already know the values of the sine and cosine of each of the two old angles θ1 and θ2.

Past experience shows that first year students, even when they have seen these identities in high school, have often not been shown where they come from geometrically, and typically find them difficult to remember reliably.

We will aim to remedy this situation by, first, showing where the sine and cosine addition formulas come from geometrically, and then providing some useful tips about how to go about memorizing these formulas in a reliable, and reproducible, way.

The following gives an outline of what is to be covered in this section.

• We first derive the sine and cosine addition formulas, showing in detail the geometry that underlies them. These formulas are very important, and will be used in a crucial way when you study how to differentiate the sine and cosine functions in your first year calculus courses.

• Having gone through the discussion of the sine and cosine addition formulas, we will see that there are a lot of steps involved in finally arriving at the final results. Even though each step is very simple, and easy to understand geometrically, the fact that there are so many steps will mean that the sine and cosine addition are things one has to memorize, since it would take too long to re-derive them from the underlying geometry each time you needed to use them again.

• To deal with this situation, we will, finally, provide some useful tips about how to go about memorizing these formulas in a way that minimizes the amount of cold memorization required, and maximizes the number of features of the formulas that can be figured out using basic information you are already know about the sine and cosine 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 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 basic geometrical definitions of the sine and cosine 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 rearranged form of these geometrical relations in which, for a right-angle triangle with non-right angle θ having hypoteneuse h, the side opposite θ of length o and the side adjacent to θ of length a, one writes o=h sin(θ) and a=h cos(θ)

• the algebraic relations giving the tangent, cotangent, secant and cosecant as ratios involving the sine and cosine (reviewable by reading (or re-reading) Section (a) 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 background material sections listed above for the first time are

• the idea and geometrical meaning of angles and angular measure (reviewable by reading (or re-reading) Chapter 1 of the course), and

• 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)

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