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Trigonometry, Geometry, Algebra and Complex Numbers
Dedicated to David Cohen (1942 - 2002)
Bruce Cohen
Lowell High School
bic@cgl.ucsf.edu
http://www.cgl.ucsf.edu/home/bic
David Sklar
San Francisco State University
dsklar46@yahoo.com
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“The shortest path between two truths in the real domain passes
through the complex domain.”
-Jacques Hadamard (1865-1963) (as quoted in Stillwell)
“The advanced reader who skips parts that appear too elementary
may miss more than the less advanced reader who skips parts that
appear too complex.”
-G Polya (as quoted in Graham Knuth and Patashnik)
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Which Truths in the Real Domain?
π 2
sin
=1=
2
2
√
√
2π π 3
3 3
· sin
=
·
=
sin
3
3
2
2
4
√
√
π 2π 3π 2
2 2 4
sin
· sin
· sin
=
·1·
= =
4
4
4
2
2
4 8
2π 3π 4π π 5
· sin
· sin
· sin
=α·β·β·α=
sin
5
5
5
5
16
...
π 2π (n − 1)π n
sin
· sin
· ... · sin
= n−1
n
n
n
2
4
Introduction
• What are our goals?
The primary goal is to look at a beautiful piece of mathematics.
We also hope to shed some light on connections between different
areas of mathematics.
• Where are we going?
A proof of the identity on the previous slide using trigonometry,
geometry, algebra, and complex numbers.
• What will we see along the way?
A proof that a regular pentagon can be constructed with a
compass and straight edge using trigonometry, geometry, algebra,
and complex numbers.
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Constructing a Regular Pentagon
cos
2π 5
= ???
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Fundamental Theorem of Algebra
• Fundamental Theorem of Arithmetic: any integer r can be
uniquely factored into (not necessarily distinct) primes:
r = p1 · p2 · ... pn
• Fundamental Theorem of Algebra: over the complex numbers any
n-th degree polynomial p(z) can be uniquely factored into n (not
necessarily distinct) linear factors:
p(z) = α(z − z1)(z − z2)...(z − zn)
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Fundamental Theorem of Algebra
• Fundamental Theorem of Algebra: over the complex numbers any
n-th degree polynomial p(z) can be uniquely factored into n (not
necessarily distinct) linear factors:
p(z) = α(z − z1)(z − z2)...(z − zn)
• Note p(z) can be thought of as an algebraic object or as a
function whose domain is the set of complex numbers.
• If we do consider a function, f (z) = p(z), then z1, z2, ..., zn form
the set of n (again not necessarily distinct) roots of the equation
p(z) = 0 (or zeros of p(z)).
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Factoring and the Fundamental Theorem
of Algebra – Examples
f1(z) = z 2 − 1 = (z − 1)(z + 1)
=⇒ zeros at 1 and − 1
f2(z) = z 2 + 1 = z 2 − (−1) = (z − i)(z + i)
=⇒ zeros at i and − i
Define i as a number whose square is − 1.
f3(z) = z 5 − 1 = (z − 1)(z 4 + z 3 + z 2 + z + 1)
=⇒ zeros at 1 and let’s dig some more
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Complex Numbers
z = a + ib
w = c + id
• Equality: z = w ⇐⇒ a = c and b = d
• Addition: z + w = (a + c) + i(b + d)
• Multiplication:
z · w = ac + iad + ibc + i2bd
z · w = ac + i(ad + bc) − bd
z · w = (ac − bd) + i(ad + bc)
• Conjugate: z = a − ib
• Absolute Value (or length): |z| =
√
zz =
√
a2 + b2
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Geometry of Complex Arithmetic
• GSP Examples
• Addition of complex numbers vector addition
• Multiplication by a real number: t(a + ib) = ta + itb
• dilationp(stretching):
p 2
2
2
|tz| = ((ta) + (tb) ) = |t|( (a + b2) = |t||z|
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Geometry of Complex Multiplication
• Multiplication by a real number:
|tz| = |t||z|
• Multiplication by i as a 90◦ rotation
• Complex Multiplication: Adding angles - mutiplying lengths
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Multiplying Polar Representations
z = r(cos θ + i sin θ)
w = s(cos φ + i sin φ)
Recall z · w = (ac − bd) + i(ad + bc)
zw = rs((cos θ cos φ − sin θ sin φ) + i(cos θ sin φ + sin θ cos φ))
= rs((cos θ cos φ − sin θ sin φ) + i(sin θ cos φ + cos θ sin φ))
The cheer: “sin cos cos sin cos cos
∴ zw = rs(cos(θ + φ) + i sin(θ + φ))
sin sin!”
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Lengths
z = r(cos θ + i sin θ)
(note r = |z|)
w = s(cos φ + i sin φ)
(s = |w|)
zw = rs(cos(θ + φ) + i sin(θ + φ))
|zw| = rs = |z||w|
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Angles and DeMoivre’s Theorem
z = r(cos θ + i sin θ)
w = s(cos φ + i sin φ)
zw = rs(cos(θ + φ) + i sin(θ + φ))
Powers of z = r(cos θ + i sin θ) :
z 2 = r2(cos 2θ + i sin 2θ)
z 3 = r3(cos 3θ + i sin 3θ)
...
z n = rn(cos nθ + i sin nθ)
n
DeMoivre’s Theorem: r(cos θ + i sin θ) = rn(cos nθ + i sin nθ)
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Using DeMoivre’s Theorem:
Find Fifth Roots of Unity
z5 = 1
(cos θ + i sin θ)5 = 1
4Π
(cos 5θ + i sin 5θ) = 1 = 1 + i0 cisA €€€€€€€€
E
5
cos 5θ = 1 and sin 5θ = 0
5θ = k · 2π
k · 2π
∴θ=
5
6Π
cisA €€€€€€€€ E
5
2Π
cisA €€€€€€€€ E
5
1
8Π
cisA €€€€€€€€ E
5
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Parametric Unit Circle
z=f(θ)=cos θ+ i sin θ
θ
0

θ

z = cos θ + i sin θ
f (θ) = cos θ + i sin θ
f 0(θ) = − sin θ + i cos θ
Show Points
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= (i2) sin θ + i cos θ
= i(i sin θ + cos θ)
= i(cos θ + i sin θ)
= if (θ)
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Rewriting f (θ)
f (θ) = cos θ + i sin θ
f 0(θ) = if (θ)
f (0) = cos 0 + i sin 0 = 1
So f satisfies a differential equation of the form:
f 0(θ) = kf (θ) =⇒ f (θ) = Dekθ
with initial condition: f (0) = 1 =⇒ 1 = Dek0
So D = 1 and f 0(θ) = if (θ) =⇒ k = i
By analogy with real valued functions:
f (θ) = eiθ
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Another Polar Notation
Our function f (θ) = eiθ leads to Euler’s:
eiθ = cos θ + i sin θ
This provides a more compact polar form, rather than
z = r(cos θ + i sin θ)
we can write
z = reiθ
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Roots of z 5 − 1 = 0
z5 = 1
(eiθ )5 = 1
ei5θ = eik2π
e4 Πi5
1
i5θ = ik · 2π
k · 2π
∴θ=
5
e2 Πi5
e6 Πi5
e8 Πi5
e2 Πi5
e4 Πi5
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Factoring z 5 − 1
1
e6 Πi5
e8 Πi5
5
z − 1 = (z − 1)(z − e
2πi
5
)(z − e
4πi
5
)(z − e
6πi
5
)(z − e
z 5 − 1 = (z − 1)(z 4 + z 3 + z 2 + z + 1)
So (z 4 + z 3 + z 2 + z + 1) =
(z − e
2πi
5
)(z − e
4πi
5
)(z − e
6πi
5
)(z − e
8πi
5
)
8πi
5
)
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Conjugates on Unit Circle
z = cos θ + i sin θ
z
z+
z
z = cos θ − i sin θ
1
= z −1
z
= cos(−θ) + i sin(−θ)
= cos(θ) − i sin(θ)

z
1
z + = z + z = 2 cos(θ)
z
Given: z 4 + z 3 + z 2 + z + 1 = 0
Find: cos
2π
5
1
1
1
1
2
z + z + 1 + + 2 = (z + 2 + 1) + z + = 0
z z
z
z
1 2
1
1
2
2
z+
= z + 2 + 2 = (z + 2 + 1) + 1
z
z
z
1
1 2
z+
−1+ z+
=0
z
z
1
1 2 + z+
−1=0
z+
z
z
2
22
(1)
(2)
(3)
(4)
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Finding cos
2π
5
1
1 2 + z+
− 1 = 0, a quadratic
We have z +
z
z
√
√
1
−1 ± 1 + 4 −1 ± 5
z+
=
=
z
2
2
1
(conjugates)
For any z of length 1, z + = 2 cos(θ)
z
2π is positive, we have:
Since cos
5
2π √5 − 1
cos
=
5
4
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Using cos
ei2А5
2π
5
2π √5 − 1
=
cos
5
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Start with a unit circle. Construct
this
√
length. (How do you get 5?) Use it
to mark off the green segment on the
x-axis. Construct the green dashed
perpendicular to get ei2π/5 , and set
the compass to the length of the
chord from 1 to ei2π/5. to mark off
the other three 5th roots.
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Which Truths in the Real Domain?
π 2
sin
=1=
2
2
√
√
2π π 3
3 3
· sin
=
·
=
sin
3
3
2
2
4
√
√
π 2π 3π 2
2 2 4
sin
· sin
· sin
=
·1·
= =
4
4
4
2
2
4 8
2π 3π 4π π 5
· sin
· sin
· sin
=α·β·β·α=
sin
5
5
5
5
16
...
π 2π (n − 1)π n
sin
· sin
· ... · sin
= n−1
n
n
n
2