Transcript Document

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
2
n
sin
 n 1
n

n
2
n1
Note that the identity
sin 
sin n sin 2n
n 1
n

n
2
n1
is equivalent to the more geometrically interesting identity
 2sin n  2sin 

2
n
 2sin
 n 1
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 nn1   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
2k 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 e
21 i n
1 e
2 2  i n
1 e
2 n 1 i n

 1 e
21 i n
 1  e    1  e 
2 2 i n
2 n 1 i n

e
2k i
n
e
2 i
n
e
2k 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  e21 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, e21 i n , e22 i n , e23 i n ,
, e2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
2k i
n
e
2 i
n
e
2k i
n
e
i
z
2 i
n
en
1
1
e
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, e21 i n , e22 i n , e23 i n , , e2n1 i n ,
hence the polynomial z n  1 factors uniquely as a product of linear factors

z n  1   z  1 z  e21 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   zn1  zn2  zn3 
 z1  1 
 z  1 g  z 
 z  1 and g 1  n . Finally we have

the product of the lengths of the chords  1  e  
2 1 i n
 1  e    1  e 
2 2 i n
2 n 1 i n
  g 1  n
The Pictures
 n 1
n
e
2k i
n
 n 1

n
e
k
n
n
2 i
n
i

2 sin ( k/n )
sin ( k/n )
k
n
e
2k 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

2k i
n
e
2 i
n
en
1
1
2  n 1 i
n
e

 z  e  
2 2 i n
2 1 i n
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 
Hence g  z   z n1  z n2  z n3 

2 n 1  i n
To evaluate g 1 , observe that 1, e21 i n , e2 2 i n , e23 i n ,

2 n 1 i n
2 2 i n
the product of the lengths of the chords 
and consider z n  1   z  1 z  e21 i n
2  n 1 i
n
1  e2 n 1 i n
    1  e    1  e 
  z  e    and note that
 1 e
z
2 i
n
1
the product of the lengths of the chords  1  e21 i n 1  e2 2 i n
Define g  z   z  e21 i n
e
i
n
e
e
2k 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