# Composite Great Circle Formula UK Set

## 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.