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
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
a 2 = b 2 + c 2 − 2bc cos A,
and understands that the art of solving a triangle involves recognition as to which formula is appropriate under which circumstances. Two quick examples - each with a warning - will suffice. Example: A plane triangle has sides a = 7 inches, b = 4 inches and angle B = 28o. Find the angle A.
See figure III.2. We use the sine formula, to obtain
sin A =
7 sin 28o = 0.821575 4
A = 55o 14'.6
3 The pitfall is that there are two values of A between 0o and 180o that satisfy sin A = 0.821575, namely 55o 14'.6 and 124o 45'.4. Figure III.3 shows that, given the original data, either of these is a valid solution.
The lesson to be learned from this is that all inverse trigonometric functions (sin-1 , cos-1 , tan-1 ) have two solutions between 0o and 360o . The function sin-1 is particularly troublesome since, for positive arguments, it has two solutions between 0o and 180o . The reader must always be on guard for "quadrant problems" (i.e. determining which quadrant the desired solution belongs to) and is warned that, unless particular care is taken in programming calculators or computers, quadrant problems are among the most frequent problems in trigonometry, and especially in spherical astronomy. Example: Find x in the triangle illustrated in figure III.4.
4 Application of the cosine rule results in 25 = x2 + 64 − 16x cos 32o Solution of the quadratic equation yields x = 4.133 or 9.435 This illustrates that the problem of "two solutions" is not confined to angles alone. Figure III.4 is drawn to scale for one of the solutions; the reader should draw the second solution to see how it is that two solutions are possible. The reader is now invited to try the following "guaranteed all different" problems by hand calculator. Some may have two real solutions. Some may have none. The reader should draw the triangles accurately, especially those that have two solutions or no solutions. It is important to develop a clear geometric understanding of trigonometric problems, and not merely to rely on the automatic calculations of a machine. Developing these critical...
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