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# 3: Plane and Spherical Trigonometry

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

• 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.
• 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.
• 3.3: Cylindrical and Spherical Coordinates
• 3.4: Velocity and Acceleration Components
• 3.5: Spherical Triangles
We are fortunate in that we have four formulas at our disposal for the solution of a spherical triangle, and, as with plane triangles, the art of solving a spherical triangle entails understanding which formula is appropriate under given circumstances. Each formula contains four elements (sides and angles), three of which, in a given problem, are assumed to be known, and the fourth is to be determined.
• 3.6: Rotation of Axes, Two Dimensions
• 3.7: Rotation of Axes, Three Dimensions. Eulerian Angles
• 3.8: Trigonometrical Formulas
A reference a set of commonly-used trigonometric formulas i sprovide. Anyone who is regularly engaged in problems in celestial mechanics or related disciplines will be familiar with most of them.

Thumbnails: If $$C$$ is acute, then $$A$$ and $$B$$ are also acute. Since $$A \le C$$, imagine that $$A$$ is in standard position in the $$xy$$-coordinate plane and that we rotate the terminal side of $$A$$ counterclockwise to the terminal side of the larger angle $$C$$. If we pick points $$(x_{1},y_{1})$$ and $$(x_{2},y_{2})$$ on the terminal sides of $$A$$ and $$C$$, respectively, so that their distance to the origin is the same number $$r$$, then we see from the picture that $$y_{1} \le y_{2}$$. Image constructed by Michael Corral (Schoolcraft College).