# The fundamental theorem of simple harmonic functions

## Abstract

The defining relation for a one-dimensional real-valued simple harmonic function designated as (pronounced “scene” and symbolized is introduced as ) It is theorized that this definition of the derivative as the function itself translated on the real line by a certain positive number . is a sufficient condition for proving that also satisfies the one-dimensional oscillator equation A proof of this hypothesis is provided by establishing the continuity, differential, and boundedness properties of . and all of its higher-order derivatives and antiderivatives, while relying critically on Roe’s characterization of the sine function. The function enables the adoption of independent non-circular definitions of the trigonometric functions *sine* and *cosine*. The properties of are investigated, and trigonometric symmetries and identities are derived directly from the defining relation and its corollaries. A formulation for the unique solution of the function is proposed. Several useful operational theorems are stated and proved. The function is applied to solve problems in trigonometry and physics.

## 1. Introduction

The importance of oscillations in science and technology cannot be overestimated. Indeed, the subject is so familiar to mathematicians, physicists, and engineers that it needs little introduction. While their origin extends back to antiquity in describing triangles and geometry, the *sine* and *cosine* functions have evolved in concept, definition, and usage. They constitute the simplest mathematical description of spatial and temporal periodic oscillations, namely the simple harmonic function. It was Leonhard Euler who first recognized that the simple harmonic motion of a forced mechanical oscillator could be described by transcendental functions of time which he defined analytically to be the *sine* and the *cosine* functions Reference 1. The properties and the identities of these functions were not necessarily determined from line segments, circles, or triangles any longer, but from the defining differential equations Reference 2. This ushered in the mathematical techniques of analytical trigonometry. The consistency between the analytically obtained and the geometrically obtained properties was convincing of the validity of the new functions, and very satisfying as well. Later, Euler advanced the functions further by establishing their close and useful relationship with the complex exponential function. Powerful techniques based on analytical trigonometry, the Fourier theorems in particular, found immense applicability in mathematics and scientific fields. The *sine* and the *cosine* functions have become deeply rooted central paradigms in mathematics.

The history of mathematics is characterized by an incremental evolution in methods and concepts. Even in recent times, the characterization of the trigonometric functions continues to advance toward completeness. A few decades ago, Roe Reference 3 had arguably provided the most important contribution to advancing our understanding of trigonometric functions in a very long time. Roe’s characterization can be summarized as follows. Any function on the real line with the property that all its derivatives and antiderivatives are uniformly bounded must be a solution to the one-dimensional oscillator equation

which is necessarily satisfied by simple harmonic oscillations with period

The present paper continues the effort toward a completeness of our understanding of trigonometric functions. Herein, the concepts of *sine* and *cosine* functions are re-evaluated through the fine-tuning of their defining relations. We show that one of the properties of the *sine* and *cosine* functions, namely their translational symmetry, can be promoted to a fundamental defining relation. In order to avoid confusion with the longstanding notions of *sine* and *cosine*, we define a new primary function *sine* and *cosine* functions can then be re-defined as translations of this function.

The main aim in the present paper is to prove that the real-valued function

satisfies the conditions in Roe’s theorem to qualify as a solution to the oscillator equation Equation 1.1. We use a slightly different formulation from that used by Roe, but eventually we recast it into Roe’s notation for comparison. We investigate the numerical and symmetry properties of the *sine* and *cosine*, and we derive trigonometric identities and properties relying only on the fundamental relation and its development, without resorting to geometry, power series expansions, or complex exponential functions. We then propose a formulation for

## 2. The theorem

## 3. Properties of the function

In the following discussion, we will interchangeably use the

### 3.1. Simple harmonic first-order differential equations

The simple harmonic function

Together with the second-order oscillator equation,

### 3.2. Relation to trigonometric functions

*sine* and *cosine*. The *sine* and *cosine* are defined as satisfying the coupled first-order equations

with boundary conditions (BC’s)

The equations are decoupled in second order such that

with BC’s

Having the foresight of conventional trigonometric functions and their properties will guide the derivation of the properties and symmetries of

### 3.3. Boundary conditions

The existence and uniqueness theorem guarantees unique solutions to linear second-order ordinary differential equations. These are determined by specifying the value of the function and its first derivative at a given point. Since the first derivative of

### 3.4. Periodicity

As derived above,

### 3.5. Translational antisymmetry

### 3.6. Equidistant extrema and zero crossings

Let the points

In other words, the argument increases by a uniform value of

### 3.7. Linear independence

The functions

### 3.8. The sum of and is constant

Taking the first derivative, of the sum of squares, we obtain

### 3.9. Arbitrary solutions to the simple harmonic differential equations

Any arbitrary function

where

### 3.10. The central theorem of trigonometry (the sum angle relation)

The relation

Taking the first and second partial derivatives of this expression with respect to