# A geometrical proof of sum of $\cos n\varphi$

## TL;DR

This paper introduces a purely geometrical proof for the sum of cosines series, providing an alternative to complex number methods and deriving a new summation formula through geometric construction.

## Contribution

It offers a novel geometrical proof for the sum of cosines series and derives an alternative summation formula, differing from existing complex number approaches.

## Key findings

- Geometrical proof of sum of cosines series
- New summation formula derived from geometric construction
- Alternative to complex number methods

## Abstract

In this article, we present a geometrical proof of sum of $\cos n\varphi$ where $n$ goes from $1$ up to $m$. Although there exist some summation forms and the proofs are simple, they use complex numbers. Our proof comes from a geometrical construction. Moreover, from this geometrical construction we obtain an other summation form.