Geometry, Trigonometry, Algebra, and Complex Numbers. Dedicated to David Cohen (1942 – 2002). David Sklar firstname.lastname@example.org. Bruce Cohen Lowell High School, SFUSD email@example.com http://www.cgl.ucsf.edu/home/bic. Palm Springs - November 2004. A Plan. A brief history.
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and Complex Numbers
Dedicated to David Cohen (1942 – 2002)
Lowell High School, SFUSD
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
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.
A Course of
Fourth edition 1927
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
Finally we have
2 sin ( kp/n )
2 sin ( kp/n )
The Short version
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
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