On sine and cosine

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

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 \(P\).

Trigonometric circle.
Trigonometric circle.

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!