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Euclidean geometry and trigonometry
Euclidean geometry means flat space
sine and cosine
y
q
Calculating 𝜋
x
Trigonometric identities
𝛽
ACME
2𝜋
𝛼
1
Euclidean geometry
(1) Line segment
A
B
(2) Extend line
segment into line
D
C
F
E
(3) Use line segment
to define circle
(4) All right
angles are equal
(5) Parallel postulate
2
Euclidean geometry: Flat space
Non-embeddable spaces
(Cannot be drawn as
rippled surfaces in higherdimensional flat spaces)
Flat
Curved
(5) Parallel postulate
3
Euclidean geometry: Pythagorean theorem
c
b
a
4
Euclidean geometry: Pythagorean theorem
Want to show
a2 + b2 = c2
c2
c
b
b2
a
a2
5
Euclidean geometry: Pythagorean theorem
Want to show
c
a2 + b2 = c2
b
(a - b)2 + 4ab/2 = c2
a2 -2ab + b2 + 2ab = c2
ab/2
a
ab/2
a2 + b2 = c2
a-b
(a – b)2
b
ab/2
ab/2
𝑐=
𝑎2 + 𝑏 2
6
Euclidean geometry and trigonometry
Euclidean geometry means flat space
sine and cosine
y
q
Calculating 𝜋
x
Trigonometric identities
𝛽
ACME
2𝜋
𝛼
7
Trigonometry: sine and cosine
y
1
q
q
x
8
Trigonometry: sine and cosine
y
1
y = sin(q)
ACME
q
x = cos(q)
x
2𝜋
9
Trigonometry: sine and cosine
y
sin 𝜃
1
3 2
2 2
1
x
2
cos 𝜃
0
𝜃
𝜋 𝜋 𝜋
6 4 3
𝜋
2
2𝜋 3𝜋 5𝜋
3 4 6
𝜋
7𝜋 5𝜋 4𝜋
6 4 3
3𝜋
2
5𝜋 7𝜋 11𝜋
2𝜋
3 4 6
−1 2
− 2 2
− 3 2
-1
10
Trigonometry: sine and cosine
sin 𝜃
1
3 2
2 2
1
2
cos 𝜃
0
𝜃
𝜋 𝜋 𝜋
6 4 3
𝜋
2
2𝜋 3𝜋 5𝜋
3 4 6
𝜋
7𝜋 5𝜋 4𝜋
6 4 3
3𝜋
2
5𝜋 7𝜋 11𝜋
2𝜋
3 4 6
−1 2
− 2 2
− 3 2
-1
11
Euclidean geometry and trigonometry
Euclidean geometry means flat space
sine and cosine
y
q
Calculating 𝜋
x
Trigonometric identities
𝛽
ACME
2𝜋
𝛼
12
Trigonometry: 𝜋
Want to approximate 𝜋
1
𝜋
3
1
𝜋
3
𝜋
3
1
𝜋
3
1
1
𝜋
3
1
𝜋
3
𝜋
3
1
ACME
𝜋
3
𝜋
3
2𝜋
13
Trigonometry: 𝜋
Want to approximate 𝜋
𝜋
3
1
1
𝜋
3
𝜋
3
1
14
Trigonometry: 𝜋
Want to approximate 𝜋
1
2
𝜋
𝜋 𝜋
3
6 6
2
+ 𝑥 2 = 12
𝑥2 = 1 −
𝑥=
1
x
3
2
1
4
3
3
=
4
2
1
𝜋
3
𝜋
3
1/2
1
1/2
15
Trigonometry: 𝜋
Want to approximate 𝜋
𝜋
6
1
2
𝜋
6
2
+ 𝑥 2 = 12
𝑥2 = 1 −
𝑥=
1
x
3
2
1
4
3
3
=
4
2
1
1
1/2
𝜋
3
𝜋
3
𝜋
6
1/2
3
2
1/2
16
Trigonometry: 𝜋
Want to approximate 𝜋
1
2
2
+ 𝑥 2 = 12
𝑥2 = 1 −
𝑥=
1
4
3
3
=
4
2
1
1/2
𝜋
6
3
2
17
Trigonometry: 𝜋
𝜋
6
1
1/2
y
1
2
2
3
+ 1−
2
1
2− 3
+
4
2
STOP
𝜋
6
3
2
Want to approximate 𝜋
3
1−
2
2
= 𝑦2
2
= 𝑦2
2 − 3 = 𝑦2
𝑦=
2− 3
𝜋
≳
6
2− 3
𝜋 ≳6 2− 3
1
𝜋 ≳ 3.1058
18
Trigonometry: 𝜋
Cosine,
ACME
2𝜋
3 . 1 4 1 5 9!
𝜋 ≳ 3.1058
19
Trigonometry: sine and cosine
sin 𝜃
1
3 2
𝜋
1° ≔
180
2 2
1
2
cos 𝜃
0
𝜃
𝜋 𝜋 𝜋
6 4 3
𝜋
2
2𝜋 3𝜋 5𝜋
3 4 6
𝜋
7𝜋 5𝜋 4𝜋
6 4 3
3𝜋
2
5𝜋 7𝜋 11𝜋
2𝜋
3 4 6
−1 2
− 2 2
− 3 2
-1
20
Euclidean geometry and trigonometry
Euclidean geometry means flat space
sine and cosine
y
q
Calculating 𝜋
x
Trigonometric identities
𝛽
ACME
2𝜋
𝛼
21
Trigonometry: sine and cosine in terms of right triangles
y
1
y = sin(q)
q
x
x = cos(q)
22
q
r cos(q)
sin(q)
1
q
cos(q)
R
R sin(q)
r
r sin(q)
Trigonometry: sine and cosine in terms of right triangles
q
R cos(q)
23
Proving identities: Pythagorean identity
STOP
cos2 𝜃 + sin2 𝜃 = ?
cos2 𝜃 + sin2 𝜃 = 1
Pythagorean identity
sin(q)
1
q
cos(q)
24
Proving identities: Angle addition formula
Want to show
sin 𝛼 + 𝛽 = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
𝛽
sin 𝛽
ℎ sin 𝛼
=
𝑥
ℎ cos 𝛼
sin 𝛼 + 𝛽
1
x
𝜋
−𝛼
2
𝛼
𝜋
−𝛽
2
𝛼+𝛽
h
25
Proving identities: Angle addition formula
Want to show
sin 𝛼 + 𝛽 = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
cos 𝛼
sin 𝛽
+ cos 𝛽 sin 𝛼 = sin 𝛼 + 𝛽
sin 𝛼
𝛽
sin 𝛼 + 𝛽
𝛼
26