# On sine and cosine

*Relating the geometric intuitions of sine and cosine, to their formal expressions in terms of Taylor series.* [preprint]

In the context of the picture below, the reader probably learned in high-school to define \(\cos \theta\) and \(\sin \theta\) as the \(x\) and \(y\) coordinates of point \(P\).

The problem is that these non-rigorous definitions, are completely detached from what one learns later (usually in undergraduate analysis), namely to define cosine and sine in terms of their Taylor series:

\[ \cos x = \sum_{n = 0}^{+\infty} \left( (-1)^n \frac{x^{2n}}{(2n)!}\right) \quad \textrm{ and } \quad \sin x = \sum_{n = 0}^{+\infty} \left((-1)^n \frac{x^{2n + 1}}{(2n + 1)!} \right) \]

This paper bridges that gap, by defining \(\cos \theta\) in terms of the *area* of the corresponding sector (in purple in the image), and from there deriving everything else: the definition of \(\sin \theta\), the Taylor expansions of both, and some usual properties of these functions (e.g. the sine and cosine of sums, etc.).

**August 28, 2023.** *Got feedback? Great, email is your friend!*

Preprint update, September 2, 2023. At the end of the conclusion, \(\pi\) was being defined as \(\int_{-1}^{1} \sqrt{1 - t^2}dt\), which is wrong. The correct definition of \(\pi\) is \(2\int_{-1}^{1} \sqrt{1 - t^2}dt\). This has now been corrected.

Preprint update, October 20, 2023. Clarified the idea underlying the definition of cosine.