PHY115: Professional Skills in Physics and Astronomy

Positional Astronomy: Session 3

Spherical Trigonometry

To make calculations with spherical triangles, we need to know the laws of spherical trigonometry, the direct analogue of the planar trigonometry you know and love. The figure below defines terminology which will be useful. A spherical triangle is formed from arcs along three great circles (dotted lines). The arc lengths are denoted by a,b,c and the vertex angles by A,B,C. The labels are chosen so that the arc 'c' is opposite vertex 'C'.
Figure 2: A great circle.
One thing to note: the arc lengths are measured as angles. For example, the distance from the north pole to the equator along a meridian can be written as 10,006 km, or as 90°, or as π/2 radians. The latter two are more useful, as they don't depend upon the radius of the sphere; for that reason it makes sense to use angles to measure both arc lengths and vertex angles of spherical triangles.
All of the problems in this section of the course can be solved with the definition of a spherical triangle, a bit of thought, and two simple rules. The difficult part of this is choosing which spherical triangle will help you with the problems. Once that's done, the two rules below will help you work out the quantity of interest.

The sine rule
The cosine rule
If you want to see how to derive these rules, there is a derivation in pages 52-54 of Astronomy: Principles and Practice by A. E. Roy and D. Clarke. A copy of this book is in the astro lab, and the main library also has copies.
We have to be a little careful about how the vertex angles are defined. Strictly, they are defined so the vertex angle B is the angle between the tangents between the two great circle arcs that make up point B, as shown in the diagram below.
Figure 3: the definition of vertex angles. Red lines mark the tangents to the great circle arcs.

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