Euclidean geometry and trigonometry

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 )