If , then is the point on the unit circle such that the length of the arc joining to (measured in the counterclockwise direction) is equal to . (There is an optical illusion in the figure. The length of segment is equal to the length of arc .)

where is the reflection about the horizontal axis. Thus if , then is the point obtained by starting at and moving along the unit circle in the clockwise direction.

**Remark**: The definition of depends on several ideas that we have not
defined
or stated assumptions about, e.g., *length of an arc* and *
counterclockwise
direction*. I believe that the amount of work it takes to formalize these
ideas at this
point is not worth the effort, so I hope your geometrical intuition will carry
you
through this chapter. (In this chapter we will assume quite a bit
of Euclidean geometry, and a few properties of area that
do not follow from our assumptions stated in chapter 5.)

A more self contained treatment of the trigonometric functions
can be found in [44, chapter 15], but the treatment given
there uses ideas that we will consider later, (e.g. derivatives,
inverse functions,
the intermediate value theorem, and the fundamental theorem of calculus)
in order to *define* the trigonometric functions.

We have the following values for :

In general

(We read ``" as `` cosine of ", and we read ``" as `` sine of ".)

Since is on the unit circle, we have

and

The equations (9.3) - (9.8) show that

and

In equation (9.2) we defined

Thus for ,

and it follows that

and consequently

Let be arbitrary real numbers. Then there exist integers and such that and . Let

Suppose (see figure). Then the length of the arc joining to is which is the same as the length of the arc joining to . Since equal arcs in a circle subtend equal chords, we have

and hence

You can verify that this same relation holds when .

Proof: From (9.10) we know

i.e.,

Hence

By expanding the squares and using the fact that for all , we conclude that

This is equation (9.13). To get equation (9.12) replace by in (9.16). If we take in equation (9.16) we get

or

If we replace by in this equation we get

Now in equation (9.16) replace by and get

or

which is equation (9.14). Finally replace by in this last equation to get (9.15).

In the process of proving the last theorem we proved the following:

Proof: We have

and

By adding these equations, we get (9.22). By subtracting the first from the second, we get (9.24).

In equation (9.24) replace by
and replace by
to get

or

This yields equation (9.25).

From the geometrical description of sine and cosine, it follows that as increases for to , increases from to and decreases from to . The identities

indicate that a reflection about the vertical line through carries the graph of sin onto the graph of cos, and vice versa.

The relation shows that

i.e., the graph of is carried onto itself by a rotation through about the origin.

We have

and , so and

(approximately).

With this information we can make a reasonable sketch of the graph of and (see the figure above)

Include an explanation for how you found
and
(or
and
). For the
remaining values
you do not need to include an explanation.

Most of the material from this section
was discussed by Claudius Ptolemy (fl.
127-151
AD). The functions considered by Ptolemy were not the sine and cosine, but the
*chord*, where the chord of an arc is the length of the segment joining
its
endpoints.

Ptolemy's chords are functions of arcs (measured in degrees), not of numbers. Ptolemy's addition law for was (roughly)

where is the diameter of the circle, and Ptolemy produced tables equivalent to tables of for in intervals of . All calculations were made to 3 sexagesimal (base ) places.

The etymology of the word *sine* is
rather curious[42, pp 615-616].
The function we call sine was first
given a name by Aryabhata near the start of the sixth century AD.
The name meant ``half chord''
and was later shortened to *jya* meaning ``chord''. The Hindu
word was translated into Arabic as *jîba*, which was a meaningless
word phonetically derived from *jya*, but (because the vowels
in Arabic were not written) was written the same as
*jaib*, which means bosom. When the Arabic was
translated into Latin it became *sinus*. (*Jaib* means bosom, bay,
or breast: *sinus* means bosom, bay or the fold of a toga around the
breast.) The English word *sine* is derived from *sinus* phonetically.