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# 9.1 Properties of Sine and Cosine

9.1   Definition () We define a function as follows.

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 .)

Thus to find , you should start at and move along the circle in a counterclockwise direction until you've traveled a distance . Since the circumference of the circle is , we see that . (Here we assume Archimedes' result that the area of a circle is half the circumference times the radius.) If , we define
 (9.2)

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 :

 (9.3) (9.4) (9.5) (9.6) (9.7)

In general

 (9.8)

9.9   Definition (Sine and cosine.) In terms of coordinates, we write

(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

In terms of components

and consequently

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

Then , so

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
 (9.10)

You can verify that this same relation holds when .

9.11   Theorem (Addition laws for sine and cosine.) For all real numbers and ,
 (9.12) (9.13) (9.14) (9.15)

Proof: From (9.10) we know

i.e.,

Hence

By expanding the squares and using the fact that for all , we conclude that
 (9.16)

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:

9.17   Theorem (Reflection law for sin and cos.) For all ,
 (9.18)

9.19   Theorem (Double angle and half angle formulas.) For all we have

9.20   Exercise. A Prove the four formulas stated in theorem 9.19.

9.21   Theorem (Products and differences of sin and cos.) For all in ,
 (9.22) (9.23) (9.24) (9.25) (9.26)

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).

9.27   Exercise. Prove equations (9.23) and (9.26).

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 condition indicates that the reflection about the vertical axis carries the graph of to itself.

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)

9.28   Exercise. Show that

9.29   Exercise. A Complete the following table of sines and cosines:

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.

 (9.30)

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.

9.31   Entertainment (Calculation of sines.) Design a computer program that will take as input a number between and , and will calculate . (I choose instead of so that you do not need to know the value of to do this.)

Next: 9.2 Calculation of Up: 9. Trigonometric Functions Previous: 9. Trigonometric Functions   Index
Ray Mayer 2007-09-07