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palm_springs_2004_3 - Geometry Trigonometry Algebra and...

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Geometry, Trigonometry, Algebra, and Complex Numbers Palm Springs - November 2004 Dedicated to David Cohen (1942 – 2002) Bruce Cohen Lowell High School, SFUSD [email protected] http://www.cgl.ucsf.edu/home/bi c David Sklar [email protected]
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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
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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.
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