On the dimension of angles and their units

TL;DR

This paper explores the implications of treating angles as quantities with their own dimension and units, proposing generalized functions that are independent of specific angle units for better consistency in scientific computations.

Contribution

It introduces a framework for angles with their own dimension and units, generalizing trigonometric and exponential functions to be unit-independent and consistent in computational applications.

Findings

Angles can be treated as quantities with their own dimension.

Generalized functions are complete and independent of angle units.

Framework improves consistency in scientific and computational contexts.

Abstract

We examine implications of angles having their own dimension, in the same sense as do lengths, masses, {\it etc.} The conventional practice in scientific applications involving trigonometric or exponential functions of angles is to assume that the argument is the numerical part of the angle when expressed in units of radians. It is also assumed that the functions are the corresponding radian-based versions. These (usually unstated) assumptions generally allow one to treat angles as if they had no dimension and no units, an approach that sometimes leads to serious difficulties. Here we consider arbitrary units for angles and the corresponding generalizations of the trigonometric and exponential functions. Such generalizations make the functions complete, that is, independent of any particular choice of unit for angles. They also provide a consistent framework for including angle units in computer algebra programs.