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Geometry, Trigonometry, Algebra, and Complex Numbers Dedicated to David Cohen (1942 – 2002) Bruce Cohen Lowell High School, SFUSD [email protected] http://www.cgl.ucsf.edu/home/bic Palm Springs - November 2004 David Sklar [email protected] 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 Notice that, without comment, the authors are assuming that the student is familiar with the following trigonometric identity: sin n sin 2n sin n 1 n 2nn1 Note that the identity sin n sin 2 n sin n1 n 2nn1 is equivalent to the more geometrically interesting identity 2sin n 2sin 2 n 2sin n1 n n k n n n 1 n n n 1 2 sin ( k/n ) n 1 sin ( k/n ) k n 1 The trigonometric identity: 2sin n 2sin 2n 2sin nn1 n is equivalent to the geometric theorem: If 2n equally spaced points are placed around a unit circle and a system of n 1 parallel chords is drawn then the product of the lengths of the chords is n. k n e 2 k i n e 2 i n n n en 1 1 2 sin ( k/n ) n 1 e i 2 n 1 i 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 2sin k n the length of the kth chord 1 e 2 k i n the product of the lengths of the n 1 chords 1 e21 i n 1 e22 i n 1 e2n1 i n 1 e21 i n 1 e22 i n 1 e 2 n 1 i n e 2 k i n e 2 i n e 2 k i n e i z 2 i n en 1 e 1 2 n 1 i n e 2 n 1 i n We introduce an arbitrary complex number z and define a function g z z e21 i n Our next task is to evaluate z e z e g 1 1 e 1 e 1 e 2 2 i n 2 n 1 i n 2 1 i n 2 2 i n 2 n 1 i n . We use a well known factoring formula, the observation that the n numbers: 1, e 21 i n ,e 2 2 i n ,e 2 3 i n , ,e 2 n 1 i n are a list of the nth roots of unity, and the Fundamental Theorem of Algebra to show that g 1 n . e 2 k i n e 2 i n e 2 k i n e i z 2 i n en 1 e 1 2 n 1 i n e 2 n 1 i n The nth roots of unity are the solutions of the equation z n 1 or z n 1 0 . By the fundamental theorem of algebra the polynomial equation z n 1 0 has exactly n roots, which we observe are 1, e21 i n , e2 2 i n , e23 i n , , e2n 1 i n , hence the polynomial z n 1 factors uniquely as a product of linear factors z n 1 z 1 z e21 i n z e z e 2 n 1 i n 2 2 i n z 1 g z Using a well known factoring formula we also have z n 1 z 1 z n 1 z n 2 z n 3 Hence g z z n1 z n2 z n3 z1 1 z 1 g z z 1 and g 1 n . Finally we have the product of the lengths of the chords 1 e21 i n 1 e22 i n 1 e 2 n 1 i n g 1 n The Pictures n 1 n e 2 k i n n 1 n e k n n 2 i n i 2 sin ( k/n ) sin ( k/n ) k n e 2 k i n n e 2 i n en 1 1 e 2 n 1 i n e 2 n 1 i n z The Short version n 1 n 2 sin ( k/n ) k n e 2 k i n e 2 i n e e i n en 1 1 e 2 k i n 1 2 n 1 i n e the product of the lengths of the chords 1 e21 i n 1 e2 2 i n z e z e 2 2 i n 2 n 1 i n the product of the lengths of the chords To evaluate g 1 , observe that 1, e21 i n , e2 2 i n , e23 i n , and consider z n 1 z 1 z e21 i n Hence g z z n1 z n2 z n3 1 e 2 n 1 i n and note that g 1 , e2 n 1 i n are the nth roots of unity z e z e 2 2 i n z n 1 z 1 z n 1 z n 2 z n 3 2 n 1 i n 1 e2 n1 i n 1 e21 i n 1 e2 2 i n Define g z z e21 i n z 2 i n 2 n 1 i n z1 1 z 1 g z z 1 g z z 1 and g 1 1n 1 1n 2 11 1 n . Cotes’ Theorem (1716) (Roger Cotes 1682 – 1716) Ck C3 C2 x P C1 O Cn Cn 1 If C1C2C3 Cn is a regular n-gon inscribed in a circle of unit radius centered at O, and P is the point on OC1 at a distance x from O, then x n 1 PC1 PC2 PCn 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 2. 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