# 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 $$P$$.

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