Euclidean geometry and trigonometry
1
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2𝜋
1 𝛼
𝛽
q x
y
Euclidean geometry
2
BA
D
C
(1) Line segment
(2) Extend line segment into line
F
E
(3) Use line segment to define circle
(4) All right angles are equal
(5) Parallel postulate
Euclidean geometry: Flat space
3
(5) Parallel postulate
Flat
Curved
Non-embeddable spaces(Cannot be drawn as rippled surfaces in higher-dimensional flat spaces)
Euclidean geometry: Pythagorean theorem
4
a
bc
Euclidean geometry: Pythagorean theorem
5
a
bcc2
a2
b2
a2 + b2 = c2Want to show
Euclidean geometry: Pythagorean theorem
6
a
bc
b
a2 + b2 = c2Want to show
(a – b)2
ab/2
ab/2
ab/2
ab/2
(a - b)2 + 4ab/2 = c2
a2 -2ab + b2 + 2ab = c2
a2 + b2 = c2
𝑐=√𝑎2+𝑏2
a - b
Euclidean geometry and trigonometry
7
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2𝜋
1 𝛼
𝛽
q x
y
Trigonometry: sine and cosine
8
1
qx
y
q
Trigonometry: sine and cosine
9
1
x = cos(q)
qx
y
y = sin(q)
2𝜋
ACME
10
2𝜋𝜃
0
-1
1
𝜋𝜋4
𝜋2
3𝜋2
12
−1 /2
√2/2√3 /2
−√3 /2−√2 /2
3𝜋4
5𝜋4
7𝜋4
𝜋6
𝜋3
2𝜋35𝜋6
7𝜋6
4𝜋3
5𝜋311𝜋6
sin (𝜃 )
cos (𝜃 )
x
y
Trigonometry: sine and cosine
-1
1
12
−1 /2
√2/2√3 /2
−√3 /2−√2 /2
sin (𝜃 )
cos (𝜃 )
11
Trigonometry: sine and cosine
2𝜋0
𝜋𝜋4
𝜋2
3𝜋2
3𝜋4
5𝜋4
7𝜋4
𝜋6
𝜋3
2𝜋35𝜋6
7𝜋6
4𝜋3
5𝜋311𝜋6
𝜃
Euclidean geometry and trigonometry
12
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2𝜋
1 𝛼
𝛽
q x
y
Trigonometry:
13
Want to approximate
𝜋3
𝜋3𝜋
3
𝜋3
𝜋3
1 1
1
1 1
𝜋3
𝜋3
𝜋3
𝜋3
11
ACME
2𝜋
1
Trigonometry:
14
Want to approximate
𝜋3
𝜋3
𝜋3
1
1
1
𝜋3
𝜋3
𝜋3
1
1
1
Trigonometry:
15
Want to approximate
𝜋6𝜋6
1/2 1/2
x
( 12 )2
+𝑥2=12
𝑥2=1− 14
𝑥=√ 34=√32
√32
𝜋3
𝜋3
1 1
Trigonometry:
16
Want to approximate
𝜋6𝜋6
1/2 1/2
x
( 12 )2
+𝑥2=12
𝑥2=1− 14
𝑥=√ 34=√32
√32
√32
1/2
1
𝜋6
Trigonometry:
17
Want to approximate
( 12 )2
+𝑥2=12
𝑥2=1− 14
𝑥=√ 34=√32
√32
1/2
1
𝜋6
Trigonometry:
18
Want to approximate
√32
1/2
1
𝜋6
𝜋6
1− √32
1
y
( 12 )2
+(1− √32 )
2
=𝑦2
14+( 2−√3
2 )2
=𝑦2
2−√3=𝑦2
𝑦=√2−√3𝜋6 ≳ √2−√3
𝜋≳6 √2−√3𝜋≳3.1058
STOP
Trigonometry:
19
𝜋≳3.1058
Sine! Sine!Cosine,
Sine!
3 . 1415 9!
ACME
2𝜋
1
-1
1
12
−1 /2
√2/2√3 /2
−√3 /2−√2 /2
sin (𝜃 )
cos (𝜃 )
20
Trigonometry: sine and cosine
1 °≔ 𝜋180
0.5240.785
1.0471.571
2.0942.356
2.6183.142
3.6653.927
4.1895.236
4.7125.498
5.7606.283
30°45°
60°90°
120°135°
150°180°
210°225°
240°300°
270°315°
330°360°
2𝜋0
𝜋𝜋4
𝜋2
3𝜋2
3𝜋4
5𝜋4
7𝜋4
𝜋6
𝜋3
2𝜋35𝜋6
7𝜋6
4𝜋3
5𝜋311𝜋6
𝜃
Euclidean geometry and trigonometry
21
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2𝜋
1 𝛼
𝛽
q x
y
Trigonometry: sine and cosine in terms of right triangles
22
1
x = cos(q)
qx
y
y = sin(q)
q
q
Trigonometry: sine and cosine in terms of right triangles
23
1
q
sin(q
)
cos(q)
r
r cos(q)
r sin
(q)
R
R sin
(q)
R cos(q)
Proving identities: Pythagorean identity
24
STOP cos2 (𝜃 )+sin2 (𝜃 )=?
cos2 (𝜃 )+sin2 (𝜃 )=1Pythagorean identity
1
q
sin(q
)
cos(q)
Proving identities: Angle addition formula
25
𝛼
Want to show
sin (𝛼+𝛽 )=sin (𝛼 ) cos (𝛽 )+cos (𝛼 ) sin (𝛽 )𝛽
1
sin ( 𝛽)
cos(𝛽 )
𝜋2 − 𝛽𝜋
2 −𝛼 𝛼+𝛽
sin(𝛼
+𝛽
)
sin (𝛽 )𝑥 =
h sin (𝛼 )hcos (𝛼 )
h
xsin( 𝛽)cos
(𝛼 )
sin(𝛼 )
Proving identities: Angle addition formula
26
𝛼
Want to show
sin (𝛼+𝛽 )=sin (𝛼 ) cos (𝛽 )+cos (𝛼 ) sin (𝛽 )𝛽
sin(𝛼
+𝛽
)
sin( 𝛽)cos
(𝛼 )
sin(𝛼 )
(sin (𝛽) cos (𝛼 )sin (𝛼 )
+cos ( 𝛽))sin (𝛼 )=¿ sin (𝛼+𝛽 )¿
cos(𝛽 )
