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## PowerPoint Slideshow about 'Geometry, Trigonometry, Algebra, and Complex Numbers' - tannar

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Geometry, Trigonometry, Algebra,

and Complex Numbers

Dedicated to David Cohen (1942 – 2002)

David Sklar

Bruce Cohen

Lowell High School, SFUSD

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

Palm Springs - November 2004

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

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.

Notice that, without comment, the authors are assuming that the

student is familiar with the following trigonometric identity:

is equivalent to the more geometrically interesting identity

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 )

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

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

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

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

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

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