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,
θ_{1} + θ_{2}
and θ_{1}  θ_{2},
provided you already know
the values of the sine and cosine of each of the two old angles
θ_{1} and θ_{2}.
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 rederive 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 rereading)
Section (c) of Chapter 2
of the course)
 the rearranged form of these geometrical relations in which,
for a rightangle triangle with nonright 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 rereading)
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 rereading)
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 rereading)
Section (b) and
Section (c)
of Chapter 3 of the course)
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