Geometry, Trigonometry, Algebra, and Complex Numbers

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Geometry, Trigonometry, Algebra, and Complex Numbers. Dedicated to David Cohen (1942 – 2002). David Sklar dsklar46@yahoo.com. Bruce Cohen Lowell High School, SFUSD bic@cgl.ucsf.edu http://www.cgl.ucsf.edu/home/bic. Palm Springs - November 2004. A Plan. A brief history.

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Geometry, Trigonometry, Algebra,

and Complex Numbers

Dedicated to David Cohen (1942 – 2002)

David Sklar

dsklar46@yahoo.com

Bruce Cohen

Lowell High School, SFUSD

bic@cgl.ucsf.edu

http://www.cgl.ucsf.edu/home/bic

Palm Springs - November 2004

A Plan

A brief history

Introduction – Trigonometry background expected of a student in

a Modern Analysis course circa 1900

A “geometric” proof of the trigonometric identity

A theorem of Roger Cotes

Bibliography

Questions

A Brief History

Some time around 1995, after needing to look up several formulas involving the gamma function, Eric Barkan and I began to develop the theory of the gamma function for ourselves using the list of formulas in chapter 6 of the Handbook of Mathematical Functions by Abramowitz and Stegun as a guide.

A few months later during a long boring meeting in Adelaide, Australia, we realized why the reflection and multiplication formulas for the gamma function were almost “obvious” and immediately began trying to turn this insight into a proof of the multiplication formula.

We made good progress for a while, but we got stuck at one point and incorrectly concluded that an odd looking trigonometric identity that we could prove from the multiplication formula was all we needed.

I called Dave Cohen who found that no one he’d talked to at UCLA had seen our trig identity, but that he found a proof in Melzak and a closely related result in Hobson

About a week later I discovered a nice geometric proof of the trig identity and later found out that in the process I’d rediscovered a theorem of Roger Cotes from 1716.

About three years later, after many interruptions and unforeseen technical difficulties,

we completed our proof of the multiplication formula.

Whittaker & Watson,

A Course of

Modern Analysis,

Fourth edition 1927

student is familiar with the following trigonometric identity:

Note that the identity

is equivalent to the more geometrically interesting identity

sin ( kp/n )

2 sin ( kp/n )

If 2n equally spaced points are placed around a unit circle and a system of parallel chords is drawn then the product of the lengths of the chords is n.

The trigonometric identity:

is equivalent to the geometric theorem:

2 sin ( kp/n )

2 sin ( kp/n )

Rearranging the chords, introducing complex numbers and using the idea that absolute value and addition of complex numbers correspond to length and addition of vectors we have

2 sin ( kp/n )

Our next task is to evaluate

Fundamental Theorem of Algebra to show that

We introduce an arbitrary complex number z and define a function

We use a well known factoring formula, the observation that the n numbers:

are a list of the nth roots of unity, and the

2 sin ( kp/n )

The nth roots of unity are the solutions of the equation

By the fundamental theorem of algebra the polynomial equation

has exactly n roots, which we observe are

hence the polynomial

factors uniquely as a product of linear factors

Using a well known factoring formula we also have

and

Hence

Finally we have

sin ( kp/n )

2 sin ( kp/n )

The Pictures

2 sin ( kp/n )

2 sin ( kp/n )

The Short version

If

is a regular n-gon inscribed in a circle of unit radius centered

at O, and P is the point on

at a distance x from O, then

Cotes’ Theorem (1716)

(Roger Cotes 1682 – 1716)

Note: Cotes did not publish a proof of his theorem, perhaps because complex numbers

were not yet considered a respectable way to prove a theorem in geometry

Bibliography

1. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover,

New York, 1965

• R. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics: a Foundation
• for Computer Science, Addison-Wesley, 1989

3. E. W. Hobson, Plane Trigonometry, 7th Ed., Cambridge University Press, 1927

4. Liang-Shin Hahn, Complex Numbers and Geometry, Mathematical Association

of America, 1994

5. Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons,

New York, 1973

5. T. Needham, Visual Complex Analysis, Oxford University Press, Oxford 1997

6. J. Stillwell, Mathematics and Its History, Springer-Verlag, New York 1989

7. E. T. Whittaker & G. N. Watson, A Course of Modern Analysis, 4th Ed.

Cambridge University Press, 1927