**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.**