From Washington to Beijing or From Annville to Cedar Fort

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
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