From Lima to Yogyakarta

In the triangle formed by the South Pole S, Lima L and Yogyakarta Y, Lima is $90°-12°{02}^{\prime }=77.97°$ from S and Yogyakarta is $90°-7°{47}^{\prime }=82.22°$. Going west from Lima, the difference in longitude is $360°-(77°{01}^{\prime }+110°{26}^{\prime })=172.55°$. This is less than 180°, so the shortest route heads west across the Pacific, and the angle at S is 172.55°.

By (1.16),

so the route is $LY=158.84°$ long, about 17 700 km.

The cosine rule for the side SY gives the angle at Lima,

which is $L=20.85°$. The angle at Yogyakarta, found the same way, is 20.57°. Both angles are less than 90°, so the point V of the route nearest to the South Pole lies between the two cities.

At V the route crosses the meridian SV at right angles, so the triangle SLV has a right angle at V and the side SL opposite it. By (1.18), $\sin SV=\sin SL\sin L$, which is $\sin 77.97°\sin 20.85°=0.3481$, so $SV=20.37°$.

V is 20.37° from the South Pole, at latitude $90°-20.37°=69.63°\text{ S}$.