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

  2. 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

  3. Euclidean geometry: Flat space Non-embeddable spaces (Cannot be drawn as rippled surfaces in higher-dimensional flat spaces) Flat Curved (5) Parallel postulate

  4. Euclidean geometry: Pythagorean theorem b c a

  5. Euclidean geometry: Pythagorean theorem Want to show a2 + b2 = c2 c2 b2 b c a a2

  6. 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

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

  8. Trigonometry: sine and cosine y 1 q q x

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

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

  11. Trigonometry: sine and cosine 1 0 -1

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

  13. Trigonometry: Want to approximate 1 ACME 1 1 1 1 1 1 1

  14. Trigonometry: Want to approximate 1 1 1

  15. Trigonometry: Want to approximate 1 1 x 1/2 1/2 1

  16. Trigonometry: Want to approximate 1 1 x 1 1/2 1/2 1/2

  17. Trigonometry: Want to approximate 1 1/2

  18. Trigonometry: Want to approximate 1 y 1/2 STOP 1

  19. Trigonometry: Sine! ACME Sine! 1 sine, Co Sine! 4 . 3 1 1 5 9!

  20. 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

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

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

  23. 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

  24. Proving identities: Pythagorean identity Pythagorean identity 1 sin(q) q cos(q) STOP

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

  26. Proving identities: Angle addition formula Want to show

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