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Geometry, Trigonometry, Algebra, and Complex Numbers

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

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  1. 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

  2. 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

  3. 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.

  4. Whittaker & Watson, A Course of Modern Analysis, Fourth edition 1927

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

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

  7. 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:

  8. 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

  9. 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

  10. 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

  11. sin ( kp/n ) 2 sin ( kp/n ) The Pictures

  12. 2 sin ( kp/n ) 2 sin ( kp/n ) The Short version

  13. 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

  14. 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

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