# Trignometry Without Geometry

## TL;DR

This paper presents a rigorous, purely analytic approach to elementary trigonometry, deriving key theorems without relying on traditional geometric constructions, suitable for advanced mathematical contexts.

## Contribution

It introduces an analytic framework for trigonometry, eliminating the need for geometric definitions and providing a more rigorous foundation for the subject.

## Key findings

- Derivation of sine and cosine functions analytically
- Reformulation of trigonometric theorems without geometry
- Comprehensive treatment of elementary trigonometry

## Abstract

The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous approach. Our purpose here is to revisit elementary trigonometry from an entirely analytic perspective. We will give a comprehensive treatment of the sine and cosine functions and will show how to derive the familiar theorems of trigonometry without reference to geometric definitions or constructions.