Section 5-8
From Washington to Beijing
      Or, in our case
From Annville to Cedar Fort
Warm-up
            Give the latitude of each location.
  1. A point on the Equator           2. The North Pole


        ...
Warm-up
            Give the latitude of each location.
  1. A point on the Equator           2. The North Pole
          ...
Warm-up
            Give the latitude of each location.
  1. A point on the Equator           2. The North Pole
          ...
Warm-up
            Give the latitude of each location.
  1. A point on the Equator           2. The North Pole
          ...
Warm-up
            Give the latitude of each location.
  1. A point on the Equator           2. The North Pole
          ...
Warm-up
            Give the latitude of each location.
  1. A point on the Equator           2. The North Pole
          ...
Great Circle:
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere

Meridian:
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere

Meridian: Also...
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere

Meridian: Also...
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere

Meridian: Also...
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere

Meridian: Also...
Great Circle: A circle within a sphere, where the center of the
   circle is also the center of the sphere

Meridian: Also...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
Example 1
  You are on the Prime Meridian at latitude 28°32’N. Matt
Mitarnowski is also on the Prime Meridian at latitude ...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
Spherical Law of Cosines
Spherical Law of Cosines

   For a spherical triangle ABC:
Spherical Law of Cosines

       For a spherical triangle ABC:

cos c = cos a cosb + sin a sin b cosC
Spherical Law of Cosines

                 For a spherical triangle ABC:

       cos c = cos a cosb + sin a sin b cosC
Her...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3
 Let’s find the shortest distance between Annville and Cedar
Fort. It will be shorter than the distance we found ...
Example 3 (continued)
Example 3 (continued)
cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°
Example 3 (continued)
cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°

           ...
Example 3 (continued)
cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°

           ...
Example 3 (continued)
cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°

           ...
Example 3 (continued)
cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°

           ...
Example 3 (continued)
cos n = cos ( 49 2 3 ° ) cos ( 49 2 3 ° ) + sin ( 49 2 3 ° ) sin ( 49 2 3 ° ) cos 35.6°

           ...
Homework
Homework



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Notes 5-8

  1. 1. Section 5-8 From Washington to Beijing Or, in our case From Annville to Cedar Fort
  2. 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. 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. 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. 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. 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. 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. 8. Great Circle:
  9. 9. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere
  10. 10. Great Circle: A circle within a sphere, where the center of the circle is also the center of the sphere Meridian:
  11. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 38. Spherical Law of Cosines
  39. 39. Spherical Law of Cosines For a spherical triangle ABC:
  40. 40. Spherical Law of Cosines For a spherical triangle ABC: cos c = cos a cosb + sin a sin b cosC
  41. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 52. Example 3 (continued)
  53. 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. 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. 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. 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. 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. 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. 59. Homework
  60. 60. Homework p. 358 # 1 - 23

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