geometry, trigonometry, algebra, and complex numbers palm springs - november 2004 dedicated to david...
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Geometry, Trigonometry, Algebra,and Complex Numbers
Palm Springs - November 2004
Dedicated to David Cohen (1942 – 2002)
Bruce CohenLowell High School, SFUSD
[email protected]://www.cgl.ucsf.edu/home/
bic
David [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:
1
122
sin sin sin n
n nn n n
Note that the identity
is equivalent to the more geometrically interesting identity
1
122
sin sin sin n
n nn n n
122sin 2sin 2sin n
n n n n
The trigonometric identity:
122sin 2sin 2sin n
n n n n
is equivalent to the geometric theorem:
sin
( k
/n )
n 1n
n
kn
1
2 si
n (
k/n
) n 1n
n
kn
1
1n 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 si
n (
k/n
)
n 1n
n
kn
1
2k ine
2 1n i
ne
2 ine
1
ine
2 sin ( k/n )
22sin the length of the kth chord 1 k i nk n e
the product of the lengths of the 1 chords n
2 1 2 2 2 1 2 1 2 2 2 11 1 1 1 1 1i n i n n i n i n i n n i ne e e e e e
We introduce an arbitrary complex number z and define a function
2k ine
2 1n i
ne
2 ine
1
ine
2 sin ( k/n )
2 1 2 2 2 1i n i n n i ng z z e z e z e
2k ine
2 1n i
ne
2 ine
1
z
2 1 2 2 2 11 1 1 1 .i n i n n i ng e e e Our next task is to evaluate
We use a well known factoring formula, the observation that the n numbers: 2 1 2 2 2 3 2 11, , , , ,i n i n i n n i ne e e e are a list of the nth roots of unity, and the
Fundamental Theorem of Algebra to show that
1 .g n
2 1 2 2 2 11 1 i n i n n i nnz z z e z e z e
2k ine
2 1n i
ne
2 ine
1
ine
2 sin ( k/n )
2k ine
2 1n i
ne
2 ine
1
z
2 1 2 2 2 3 2 11, , , , , ,i n i n i n n i ne e e e
The nth roots of unity are the solutions of the equation
1 or 1 0 .n nz z By the fundamental theorem of algebra the polynomial equation
1 0nz has exactly n roots, which we observe are
hence the polynomial
1nz factors uniquely as a product of linear factors 1z g z
Using a well known factoring formula we also have 1 2 3 11 1 1n n n nz z z z z z 1z g z
Hence
1 2 3 1n n ng z z z z z and 1 .g n
2 1 2 2 2 1the product of the lengths of the chords 1 1 1i n i n n i ne e e 1g n
Finally we have
sin
( k
/n )
n 1n
n
kn
2 si
n (
k/n
) n 1n
n
kn
2k ine
2 1n i
ne
2 ine
1
z2k ine
2 1n i
ne
2 ine
1
ine
The Pictures
2k ine
2 1n i
ne
2 ine
1
ine
2 sin ( k/n )
2 k ine
2 1n i
ne
2 ine
1
z
2 si
n (
k/n
)
n 1n
n
kn
1
2 1 2 2 2 1and consider 1 1 i n i n n i nnz z z e z e z e
1 2 3 11 1 1n n n nz z z z z z 1z g z
1 2 3Hence 1n n ng z z z z z 1 2 1and 1 1 1 1 1 .n ng n
the product of the lengths of the chords
2 1 2 2 2 1Define and note thati n i n n i ng z z e z e z e
2 1 2 2 2 11 1 1i n i n n i ne e e
2 1 2 2 2 11 1 1i n i n n i ne e e
the product of the lengths of the chords 1g
2 1 2 2 2 3 2 1To evaluate 1 , observe that 1, , , , , are the nth roots of unityi n i n i n n i ng e e e e
1z g z
The Short version
If 1 2 3 nC C C C
1 21nnx PC PC PC
is a regular n-gon inscribed in a circle of unit radius centeredat O, and P is the point on 1OC at a distance x from O, then
Cotes’ Theorem (1716)
1C
kC3C
O
2C
nC
1nC
Px
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 1997
7. E. T. Whittaker & G. N. Watson, A Course of Modern Analysis, 4th Ed. Cambridge University Press, 1927
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
5. Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons, New York, 1973
6. J. Stillwell, Mathematics and Its History, Springer-Verlag, New York 1989
4. Liang-Shin Hahn, Complex Numbers and Geometry, Mathematical Association of America, 1994
2. R. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics: a Foundation for Computer Science, Addison-Wesley, 1989