Euclidean geometry and trigonometry
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Euclidean geometry and trigonometry. y. Euclidean geometry means flat space. sine and cosine. ACME. 1. q. x. Calculating. Trigonometric identities. Euclidean geometry. ( 1 ) Line segment. A. B. ( 2 ) Extend line segment into line. D. C. F. E. ( 5 ) Parallel postulate.

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Euclidean geometry and trigonometry

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Euclidean geometry and trigonometry

Euclidean geometry and trigonometry

y

Euclidean geometry means flat space

sine and cosine

ACME

1

q

x

Calculating

Trigonometric identities


Euclidean geometry and trigonometry

Euclidean geometry

(1) Line segment

A

B

(2) Extend line segment into line

D

C

F

E

(5) Parallel postulate

(4) All right angles are equal

(3) Use line segment to define circle


Euclidean geometry and trigonometry

Euclidean geometry: Flat space

Non-embeddable spaces

(Cannot be drawn as rippled surfaces in higher-dimensional flat spaces)

Flat

Curved

(5) Parallel postulate


Euclidean geometry and trigonometry

Euclidean geometry: Pythagorean theorem

b

c

a


Euclidean geometry and trigonometry

Euclidean geometry: Pythagorean theorem

Want to show

a2 + b2 = c2

c2

b2

b

c

a

a2


Euclidean geometry and trigonometry

Euclidean geometry: Pythagorean theorem

Want to show

a2 + b2 = c2

c

b

(a - b)2 + 4ab/2 = c2

ab/2

a2 -2ab + b2 + 2ab = c2

a

ab/2

a2 + b2 = c2

a- b

b

(a– b)2

ab/2

ab/2


Euclidean geometry and trigonometry

Euclidean geometry and trigonometry

y

Euclidean geometry means flat space

sine and cosine

ACME

1

q

x

Calculating

Trigonometric identities


Euclidean geometry and trigonometry

Trigonometry: sine and cosine

y

1

q

q

x


Euclidean geometry and trigonometry

Trigonometry: sine and cosine

y

ACME

1

y = sin(q)

q

x

x = cos(q)


Euclidean geometry and trigonometry

Trigonometry: sine and cosine

y

1

x

0

-1


Euclidean geometry and trigonometry

Trigonometry: sine and cosine

1

0

-1


Euclidean geometry and trigonometry

Euclidean geometry and trigonometry

y

Euclidean geometry means flat space

sine and cosine

ACME

1

q

x

Calculating

Trigonometric identities


Euclidean geometry and trigonometry

Trigonometry:

Want to approximate

1

ACME

1

1

1

1

1

1

1


Euclidean geometry and trigonometry

Trigonometry:

Want to approximate

1

1

1


Euclidean geometry and trigonometry

Trigonometry:

Want to approximate

1

1

x

1/2

1/2

1


Euclidean geometry and trigonometry

Trigonometry:

Want to approximate

1

1

x

1

1/2

1/2

1/2


Euclidean geometry and trigonometry

Trigonometry:

Want to approximate

1

1/2


Euclidean geometry and trigonometry

Trigonometry:

Want to approximate

1

y

1/2

STOP

1


Euclidean geometry and trigonometry

Trigonometry:

Sine!

ACME

Sine!

1

sine,

Co

Sine!

4

.

3

1

1

5

9!


Euclidean geometry and trigonometry

Trigonometry: sine and cosine

1

0

30°

45°

60°

90°

0.524

0.785

1.047

1.571

2.094

120°

2.356

135°

2.618

150°

3.142

180°

3.665

210°

3.927

225°

4.189

240°

4.712

270°

5.236

300°

5.498

315°

5.760

330°

6.283

360°

-1


Euclidean geometry and trigonometry

Euclidean geometry and trigonometry

y

Euclidean geometry means flat space

sine and cosine

ACME

1

q

x

Calculating

Trigonometric identities


Euclidean geometry and trigonometry

Trigonometry: sine and cosine in terms of right triangles

y

1

y = sin(q)

q

x

x = cos(q)


Euclidean geometry and trigonometry

Trigonometry: sine and cosine in terms of right triangles

q

r

1

R

r sin(q)

sin(q)

Rsin(q)

q

rcos(q)

Rcos(q)

cos(q)

q


Euclidean geometry and trigonometry

Proving identities: Pythagorean identity

Pythagorean identity

1

sin(q)

q

cos(q)

STOP


Euclidean geometry and trigonometry

Proving identities: Angle addition formula

Want to show

1

x

h


Euclidean geometry and trigonometry

Proving identities: Angle addition formula

Want to show


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