Euclidean geometry and trigonometry Euclidean geometry means flat

Euclidean geometry and trigonometry Euclidean geometry means flat space sine and cosine y q x Trigonometric identities ACME 1 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 a 2 + b 2 = c 2 c b b 2 a a 2 5

Euclidean geometry: Pythagorean theorem Want to show c a 2 + b 2 = c 2 b (a - b)2 + 4 ab/2 = c 2 a 2 -2 ab + b 2 + 2 ab = c 2 ab/2 a 2 + b 2 = c 2 a-b (a – b)2 b ab/2 6

Euclidean geometry and trigonometry Euclidean geometry means flat space sine and cosine y q x Trigonometric identities ACME 1 7

Trigonometry: sine and cosine y 1 q q x 8

Trigonometry: sine and cosine y 1 y = sin(q) x ACME q x = cos(q) 9

Trigonometry: sine and cosine y 1 x 0 -1 10

Trigonometry: sine and cosine 1 0 -1 11

Euclidean geometry and trigonometry Euclidean geometry means flat space sine and cosine y q x Trigonometric identities ACME 1 12

1 1 1 ACME 1 1 13

1 14

1 x 1 1/2 15

1 x 1 1 1/2 16

1 1/2 17

1 y 1/2 STOP 1 18

Sine! Si ne! Cosine, ACME ! e n i S 3. 1 4 1 5 9! 1 19

-1 2. 4 35 2. 6 61 8 3. 14 2 3. 66 3. 5 92 4. 7 18 9 4. 71 2 5. 23 6 5. 49 5. 8 76 0 6. 28 3 0° 13 5 15 ° 0° 18 0° 21 0° 22 5° 24 0° 27 0° 30 0° 31 5 33 ° 0° 36 0° 09 2. 1 ° 12 57 1. 0 90 30 ° 45 ° 60 ° 0. 52 0. 4 78 1. 5 04 7 Trigonometry: sine and cosine 1 20

Euclidean geometry and trigonometry Euclidean geometry means flat space sine and cosine y q x Trigonometric identities ACME 1 21

Trigonometry: sine and cosine in terms of right triangles y 1 y = sin(q) q x x = cos(q) 22

q 1 sin(q) r cos(q) 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 Pythagorean identity sin(q) 1 q cos(q) 24

Proving identities: Angle addition formula Want to show 1 x h 25

Proving identities: Angle addition formula Want to show 26