Astronomy Principles and Practice Fourth Edition.pdf

56 The geometry of the sphere
Figure 7.7. Example 7.5—flight route from Prestwick to Gander.
As an example take the cosine formula, namely
cos a = cos b cos c + sin b sin c cos A.
If θ is any angle, in circular measure,
cos θ = 1 − θ 2
2! + θ 4
4! −···
sin θ = θ − θ 3
3! + θ 5
−··· . (7.15)
5!
If θ is sufficiently small, we may write
cos θ = 1 − θ 2 /2 sinθ = θ.
Now let the spherical triangle ABC shrink so that the cosine formula may be written as
( )( )
1 − a2
2 = 1 − b2
1 − c2
+ b.c. cos A
2 2
or
1 − a2
2 = 1 − b2
2 − c2
2 + b2 c 2
+ b.c. cos A.
4
We can neglect b 2 c 2 /4 since it is of smaller order than a 2 /2, etc.
Hence,
a 2 = b 2 + c 2 − 2bc cos A
the well-known formula of plane trigonometry.
Example 7.5. A plane, moving with a speed of 500 knots, flies the great circle route from Prestwick
(4 ◦ 36 ′ W, 55 ◦ 30 ′ N) to Gander (54 ◦ 24 ′ W, 48 ◦ 34 ′ N). What distance does it cover and how long does
the flight take
Let A and B represent Prestwick and Gander respectively in figure 7.7 so that the great circle arc
AB is the flight route. If the meridians PBD and PAC are drawn from the north pole P through B
and A to the equator DC, triangle PBA is a spherical triangle. Applying the cosine formula, we may
write
cos AB = cos AP cos BP + sin AP sin BPcos APB
BP = 90 ◦ − 48 ◦ 34 ′ = 41 ◦ 26 ′ = 41·◦4333
AP = 90 ◦ − 55 ◦ 30 ′ = 34 ◦ 30 ′ = 34·◦5000
APB = 54 ◦ 24 ′ − 4 ◦ 36 ′ = 49 ◦ 48 ′ = 49·◦8000.