Plane and Spherical Trigonometry
1 CHAPTER 3 PLANE AND SPHERICAL TRIGONOMETRY 3.1 Introduction It is assumed in this chapter that readers are familiar with the usual elementary formulas encountered in introductory trigonometry. We start the chapter with a brief review of the solution of a plane triangle. While most of this will be familiar to readers, it is suggested that it be not skipped over entirely, because the examples in it contain some cautionary notes concerning hidden pitfalls. This is followed by a quick review of spherical coordinates and direction cosines in threedimensional geometry. The formulas for the velocity and acceleration components in twodimensional polar coordinates and three-dimensional spherical coordinates are developed in section 3.4. Section 3.5 deals with the trigonometric formulas for solving spherical triangles. This is a fairly long section, and it will be essential reading for those who are contemplating making a start on celestial mechanics. Section 3.6 deals with the rotation of axes in two and three dimensions, including Eulerian angles and the rotation matrix of direction cosines. Finally, in section 3.7, a number of commonly encountered trigonometric formulas are gathered for reference. 3.2 Plane Triangles. This section is to serve as a brief reminder of how to solve a plane triangle. While there may be a temptation to pass rapidly over this section, it does contain a warning that will become even more pertinent in the section on spherical triangles. Conventionally, a plane triangle is described by its three angles A, B, C and three sides a, b, c, with a being opposite to A, b opposite to B, and c opposite to C. See figure III.1.
B FIGURE III.1 c C b A a
2
It is assumed that the reader is familiar with the sine and cosine formulas for the solution of the triangle: a b c = = sin A sin B sin C
3.2.1
and
a 2 = b 2 + c 2 − 2bc cos A,
3.2.2
and understands that the art of solving a triangle involves recognition as to which formula is
B FIGURE III.1 c C b A a
2
It is assumed that the reader is familiar with the sine and cosine formulas for the solution of the triangle: a b c = = sin A sin B sin C
3.2.1
and
a 2 = b 2 + c 2 − 2bc cos A,
3.2.2
and understands that the art of solving a triangle involves recognition as to which formula is