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A NOTE ON THE HISTORY OF TRIGONOMETRIC FUNCTIONS AND SUBSTITUTIONS
Jean-Pierre Merlet
INRIA, BP 93, 06902 Sophia-Antipolis Cedex
France
Abstract: Trigonometric functions appear very frequently in mechanism kinematic equations (for example as soon a revolute joint is involved in the mechanism). Dealing with these functions is difficult and trigonometric substitutions are used to transform them into algebraic terms that can be handled more easily. We present briefly the origin of the trigonometric functions and of these substitutions.
Keywords: trigonometric substitutions, robotics
1 INTRODUCTION
When dealing with the kinematic equations of a mechanism involving revolute joints appears frequently terms involving the sine and cosine of the revolute joint angles. Hence when solving either the direct or inverse kinematic equations one has to deal with such terms. But the problem is that the geometric structure of the manifold build on such terms is not known and that only numerical method can be used to solve such equations. A classical approach to deal with this problem is to transform these terms into algebraic terms using the following substitution:
\[
t=\tan \left(\frac{\theta}{2}\right) \Rightarrow \sin (\theta)=\frac{2 t}{1+t^{2}} \quad \cos (\theta)=\frac{1-t^{2}}{1+t^{2}}
\]
This substitution allows one to transform an equation involving the sine and cosine of an angle into an algebraic equation. Such type of equation has a structure that has been well studied in the past and for which a large number properties has been established, such as the topology of the resulting manifold and its geometrical properties. Furthermore efficient formal/numerical methods are available for solving systems of algebraic equations.
But when asking in the community about the possible origin of this substitution I get only vague answer, most of them mentioning ancient Greek mathematicians. Hence the purpose of this paper is to investigate the origin of these substitutions.