Notes 5-8

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From Washington to Beijing or From Annville to Cedar Fort

From Washington to Beijing or From Annville to Cedar Fort

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  • 1. Section 5-8 From Washington to Beijing Or, in our case From Annville to Cedar Fort
  • 2. Warm-up Give the latitude of each location. 1. A point on the Equator 2. The North Pole 3. The South Pole 4. A place halfway between the Equator and the North Pole 5. A place halfway between the Equator and the South Pole
  • 3. Warm-up Give the latitude of each location. 1. A point on the Equator 2. The North Pole 0° 3. The South Pole 4. A place halfway between the Equator and the North Pole 5. A place halfway between the Equator and the South Pole
  • 4. Warm-up Give the latitude of each location. 1. A point on the Equator 2. The North Pole 0° 90°N 3. The South Pole 4. A place halfway between the Equator and the North Pole 5. A place halfway between the Equator and the South Pole
  • 5. Warm-up Give the latitude of each location. 1. A point on the Equator 2. The North Pole 0° 90°N 3. The South Pole 90°S 4. A place halfway between the Equator and the North Pole 5. A place halfway between the Equator and the South Pole
  • 6. Warm-up Give the latitude of each location. 1. A point on the Equator 2. The North Pole 0° 90°N 3. The South Pole 90°S 4. A place halfway between the Equator and the North Pole 45°N 5. A place halfway between the Equator and the South Pole
  • 7. Warm-up Give the latitude of each location. 1. A point on the Equator 2. The North Pole 0° 90°N 3. The South Pole 90°S 4. A place halfway between the Equator and the North Pole 45°N 5. A place halfway between the Equator and the South Pole 45°S
  • 8. Great Circle:
  • 9. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere
  • 10. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian:
  • 11. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian: Also known as longitude; A semicircle whose endpoints are the North and South Poles
  • 12. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian: Also known as longitude; A semicircle whose endpoints are the North and South Poles Prime Meridian:
  • 13. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian: Also known as longitude; A semicircle whose endpoints are the North and South Poles Prime Meridian: Also known as the Greenwich Meridian; The meridian at 0° longitude, which runs through Greenwich, England
  • 14. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian: Also known as longitude; A semicircle whose endpoints are the North and South Poles Prime Meridian: Also known as the Greenwich Meridian; The meridian at 0° longitude, which runs through Greenwich, England International Date Line:
  • 15. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian: Also known as longitude; A semicircle whose endpoints are the North and South Poles Prime Meridian: Also known as the Greenwich Meridian; The meridian at 0° longitude, which runs through Greenwich, England International Date Line: Located at 180°W and 180°E longitude; Also where the date changes from one day to the next
  • 16. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt?
  • 17. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth:
  • 18. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth: 3960 miles
  • 19. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth: 3960 miles 47°12 '− 28°32 '
  • 20. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth: 3960 miles 47°12 '− 28°32 ' = 18°40 '
  • 21. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth: 3960 miles 47°12 '− 28°32 ' = 18°40 ' = 18 2 3 °
  • 22. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth: 3960 miles 47°12 '− 28°32 ' = 18°40 ' = 18 2 3 ° 18 3 ° 2 • π • 3960 180°
  • 23. Example 1 You are on the Prime Meridian at latitude 28°32’N. Matt Mitarnowski is also on the Prime Meridian at latitude 47°12’N. What is the distance from you to Matt? Radius of Earth: 3960 miles 47°12 '− 28°32 ' = 18°40 ' = 18 2 3 ° 18 3 ° 2 • π • 3960 ≈ 1290mi 180°
  • 24. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you?
  • 25. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51°
  • 26. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6°
  • 27. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 '
  • 28. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle.
  • 29. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. L C
  • 30. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. L 3960 C
  • 31. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. L r 3960 C
  • 32. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. L r 3960 C m∠L = 40°20 '
  • 33. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. r L r cos ( 40 1 3 ° ) = 3960 3960 C m∠L = 40°20 '
  • 34. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. r L r cos ( 40 1 3 ° ) = 3960 cos ( 40 1 3 ° ) = r 3960 3960 C m∠L = 40°20 '
  • 35. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. r L r cos ( 40 1 3 ° ) = 3960 cos ( 40 1 3 ° ) = r 3960 3960 r ≈ 3019mi C m∠L = 40°20 '
  • 36. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. r L r cos ( 40 1 3 ° ) = 3960 cos ( 40 1 3 ° ) = r 3960 3960 r ≈ 3019mi C 35.6° m∠L = 40°20 ' • π • 3019 180°
  • 37. Example 2 Again, you and Matt are in two different spots on Earth. This time, you’re both at latitude 40°20’N (Annville’s latitude). You are in Annville (longitude 76.51°W, and Matt is in Cedar Fort, Utah (longitude 112.11°W). How far along the latitude is Matt from you? 112.11° − 76.51° = 35.6° = 35°36 ' This is not a great circle, so we need the radius of this circle. r L r cos ( 40 1 3 ° ) = 3960 cos ( 40 1 3 ° ) = r 3960 3960 r ≈ 3019mi C 35.6° m∠L = 40°20 ' • π • 3019 ≈ 1876mi 180°
  • 38. Spherical Law of Cosines
  • 39. Spherical Law of Cosines For a spherical triangle ABC:
  • 40. Spherical Law of Cosines For a spherical triangle ABC: cos c = cos a cosb + sin a sin b cosC
  • 41. Spherical Law of Cosines For a spherical triangle ABC: cos c = cos a cosb + sin a sin b cosC Here, a and b are sides of a spherical triangle, which is made up of arcs instead of line segments. These arcs are measured in degrees.
  • 42. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle.
  • 43. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N C A
  • 44. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° C A
  • 45. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations
  • 46. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations 90° − 40°20 '
  • 47. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations 90° − 40°20 ' = 49 2 3 °
  • 48. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations 90° − 40°20 ' = 49 2 3 ° = a
  • 49. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations 90° − 40°20 ' = 49 2 3 ° = a =c
  • 50. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations 90° − 40°20 ' = 49 2 3 ° = a =c cos n = cos a cos c + sin a sin c cos N
  • 51. Example 3 Let’s find the shortest distance between Annville and Cedar Fort. It will be shorter than the distance we found in Example 2 since the latitude they sit on is not a great circle. N m∠ANC = 35.6° Find the other two angles (and thus, the arcs) by finding the difference from the C A North Pole to the latitude o the locations 90° − 40°20 ' = 49 2 3 ° = a =c cos n = cos a cos c + sin a sin c cos N cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°
  • 52. Example 3 (continued)
  • 53. Example 3 (continued) cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°
  • 54. Example 3 (continued) cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6° cos n ≈ .8913949153
  • 55. Example 3 (continued) cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6° cos n ≈ .8913949153 cos −1 ( cos n ) ≈ cos (.8913949153) −1
  • 56. Example 3 (continued) cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6° cos n ≈ .8913949153 cos −1 ( cos n ) ≈ cos (.8913949153) −1 n ≈ 26.95°
  • 57. Example 3 (continued) cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6° cos n ≈ .8913949153 cos −1 ( cos n ) ≈ cos (.8913949153) −1 n ≈ 26.95° 26.95° • π • 3960 180°
  • 58. Example 3 (continued) cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6° cos n ≈ .8913949153 cos −1 ( cos n ) ≈ cos (.8913949153) −1 n ≈ 26.95° 26.95° • π • 3960 ≈ 1863mi 180°
  • 59. Homework
  • 60. Homework p. 358 # 1 - 23