Composite Great Circle Formula UK Set
A Composite great circle track between Montevideo (34° 35′ S 056° 10’W) and Cape town (33° 55′ S 18°25’E) is required with a limiting latitude of 38°. Find the initial course and the total distance to the steam.
Composite Great Circle Formula UK Set
Dlong = 74° 35′ E
In triangle PAV1, V1 = 90°
PA = 90° – 34° 35′ = 55° 25′
PV1 = 90° – 38° = 52°
By using Napiers rule,
sin (90 – P1) = tan (90-PA). tan (PV1)
cosP1 = tan 34° 35′ . tan 52°
P1 = 28.06°
Again, sinAV1 = cos(90-PA) . cos(90 – P1) = cos (90 – 55°25′) . sin P1
= cos 34°35′ . sin 28.06°
AV1 = 22.78° = 1367.1 miles
In triangle PBV2, V2 = 90°
PB = 90 – 33° 55′ = 56° 05′
PV2 = 90° – 38° = 52°
By using Napiers Rule,
sin (90 – P3) = tan (90-PB). tan (PV2)
cos P3 = tan 33° 55′ . tan 52°
P3 = 30.61°
sinBV2 = cos(90-PB) . cos(90 – P3) = cos 33° 55′ . sin P3 = cos 33°55′ . sin 30.61°
BV2 = 24.99° = 1499.4 miles
Now, P2 = dlong – (P1+P3)
P2 = 15.91°
We know, distance (departure) = dlong P2 * cosm’lat
= 15.91° * cos 38°
V1V2 = 752.2 miles
Total Distance = 1367.1 + 1499.4 + 752.2 = 3619.1 miles (Ans.)
Initial course, sin (90-A) = tan AV1 . tan 90-PA
cos A = tan 22.78° . tan 34°35′
So, initial course A = S 73.2° E = 106.8° (T) ANS.