KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

CHAPTER 5, SECTION F: OTHER TRIG IDENTITIES EASILY DERIVED (IN A FEW LINES AT MOST) FROM THE SINE AND COSINE ADDITION FORMULAS

OVERVIEW:

This section of Chapter 5 deals with a number of trig identities which follow in a few lines from the sine and cosine addition formulas, discussed in detail in the previous section, Section (e), of this chapter. These new identities include a number of results which some students have seen in high school, such as the half-angle formula, the double-angle formula, and the tangent and cotangent addition formulas.

Past experience shows that many first year students, even when they have seen these identities in high school, have not been made aware of how closely these new results are related to the sine and cosine addition formulas, nor how easily they can be re-derived on the spot, starting from the addition formulas. (Many students, in fact, have been required to memorize each of the new identities as as a separate fact, an approach which typically makes it very hard to remember these formulas reliably over time.)

We will aim to remedy this situation by showing how, once you have learned the sine and cosine addition formulas (as explained in Section (e) of this chapter) a lot of the other formulas you may have been given to memorize separately in high school actually follow so easily from the sine and cosine addition formulas that they do not have to be memorized at all.

The following gives an outline of the additional identities, all following from the sine and cosine addition formulas, which will be covered in this section.

• The addition formulas for the tangent and cotangent (which some students will have seen in high school), as well as addition formulas for the secant and cosecant (which very few students have seen in high school)

• The double-angle formulas (which, in fact, are just special cases of the addition formulas, and hence, even more so than the other results, REALLY don't have to be memorized at all);

• The half-angle formulas (which are just rearranged versions of the double-angle formulas);

• Some additional identities (which a small number of students have seen in high school), showing how to write any product of two sine functions, two cosine functions, or one sine and one cosine function as a sum or difference of two terms, each of which is a single sine or cosine.

In each case we will see that only a few steps are needed to derive the new result, so, as claimed above, none of these additional identities have to be memorized separately at all.

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.

For this section of the course, you should be familiar with with

• 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)

• The basic Pythagoras' Theorem based identity sin2(θ) + cos2(θ) = 1 (reviewable by reading (or re-reading) Section (b) of Chapter 5 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 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 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)