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# 10.3 Trigonometric Functions

10.45   Example. Suppose that there are real valued functions on such that

You have seen such functions in your previous calculus course. Let . Then

Hence, is constant on , and since , we have

In particular,

Let . By the power rule and chain rule,

Hence is constant and since , we conclude that for all . Since a sum of squares in is zero only when each summand is zero, we conclude that

Let

Then for all and . I will now construct a sequence of functions on such that for all , and for all . I have

It should be clear how this pattern continues. Since , is increasing on and since , for . Since on , is increasing on and since , for .

This argument continues (I'll omit the inductions), and I conclude that for all and all . Now

For each , , define

The equations above suggest that for all , ,
 (10.46)

and
 (10.47)

I will not write down the induction proof for this because I believe that it is clear from the examples how the proof goes, but the notation becomes complicated.

Since , and , the relation (10.46) actually holds for all (not just for ) and similarly relation (10.47) holds for all . From (10.46) and (10.47), we see that if is a null sequence, then the sequence converges to , and if is a null sequence, then converges to .

We will show later that both sequences and converge for all complex numbers , and we will define

 (10.48) (10.49)

for all . The discussion above is supposed to convince you that for real this definition agrees with whatever definition of sine and cosine you are familiar with. The figures show graphs of and for small .

Graphs of the polynomials for

Graphs of the polynomials for

10.50   Exercise. A Show that and are null sequences for all complex with .

10.51   Exercise. A a) Using calculator arithmetic, calculate the limits of
and accurate to 8 decimals. Compare your results with your calculator's value of and . [Be sure to use radian mode.]

b) Calculate to 3 or 4 decimals accuracy. Note that is real.

The figure shows graphical representations for , , , and . Note that is the identity function.

10.52   Entertainment. Show that for all

and

Use a trick similar to the trick used to show that and .

10.53   Entertainment. By using the definitions (10.48) and (10.49), show that

a) For all , is real, and .

b) For all , is pure imaginary, and if and only if .

c) Assuming that the identity

is valid for all complex numbers and , show that if then sin maps the horizontal line to the ellipse having the equation

d) Describe where maps vertical lines. (Assume that the identity holds for all

10.54   Note. Rolle's theorem is named after Michel Rolle (1652-1719). An English translation of Rolle's original statement and proof can be found in [46, pages 253-260]. It takes a considerable effort to see any relation between what Rolle says, and what our form of his theorem says.

The series representations for sine and cosine (10.48) and (10.49) are usually credited to Newton, who discovered them some time around 1669. However, they were known in India centuries before this. Several sixteenth century Indian writers quote the formulas and attribute them to Madhava of Sangamagramma (c. 1340-1425)[30, p 294].

The method used for finding the series for sine and cosine appears in the 1941 book What is Mathematics" by Courant and Robbins[17, page 474]. I expect that the method was well known at that time.

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