Derivation of Formulas in Spherical Trigonometry Based on Rotation Matrix

Abstract

The formulas of spherical triangle, which are widely used to solve various navigation problems, are the important basic knowledge of nautical mathematics. Because the sine rules and the cosine rules for the sides are the fundamental formulas to derive the other spherical triangle formulas, they are also called the genetic codes of the spherical triangle formulas. In the teaching process, teachers usually use the geometric method to derive and prove these fundamental formulas. However, the derivation of geometric methods is complicated and difficult to understand. To improve the teaching process, this paper proposes the three-dimensional rotation method, which is based on conversion of two cartesian coordinate frames using the rotation matrices. This method can easily and simultaneously derive the sine rules, the cosine rules for the sides, and the five-part formulas (I), and is also helpful to solve different kinds of spherical navigation problems.

Keywords

• Spherical Trigonometry
• Rotation Matrix
• Spherical Triangle
• Formulas of Spherical Triangle
• Nautical Mathematics
• Geometric Method
• Spherical Navigation
• Astronavigation