Geometry, Trigonometry, Algebra, and Complex Numbers Palm Springs - November 2004 Dedicated to David Cohen (1942 – 2002) Bruce Cohen Lowell High School, SFUSD David Sklar

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 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

Notice that, without comment, the authors are assuming that the student is familiar with the following trigonometric identity:

Note that the identity is equivalent to the more geometrically interesting identity

The trigonometric identity: is equivalent to the geometric theorem: sin ( k  /n ) 2 sin ( k  /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.

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 ( k  /n )

We introduce an arbitrary complex number z and define a function 2 sin ( k  /n ) Our next task is to evaluate We use a well known factoring formula, the observation that the n numbers: are a list of the nth roots of unity, and the Fundamental Theorem of Algebra to show that

2 sin ( k  /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 Hence and Finally we have

sin ( k  /n ) 2 sin ( k  /n ) The Pictures

2 sin ( k  /n ) The Short version

Ifis 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) 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 (Roger Cotes 1682 – 1716)

Bibliography 5. T. Needham, Visual Complex Analysis, Oxford University Press, Oxford E. T. Whittaker & G. N. Watson, A Course of Modern Analysis, 4th Ed. Cambridge University Press, M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, E. W. Hobson, Plane Trigonometry, 7 th Ed., Cambridge University Press, Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons, New York, J. Stillwell, Mathematics and Its History, Springer-Verlag, New York Liang-Shin Hahn, Complex Numbers and Geometry, Mathematical Association of America, R. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics: a Foundation for Computer Science, Addison-Wesley, 1989