On sine and cosine

Relating the geometric intuitions of sine and cosine, to their formal expressions in terms of Taylor series.
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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, February 21, 2024. Big overhaul of the paper.

• Function $$A$$, the basis for defining cosine formally, is now first proved to be a bijection, which simplifies explaining why it is suited to be used to define cosine.

• Extending sine and cosine from $$[0, 2\pi]$$ to all of $$\mathbb{R}$$ is done in a way that no longer overlaps at $$2\pi$$.

• This in turn aids in explicitly proving that the derivation rules for both functions, in $$]0, 2\pi[$$, also apply to $$\mathbb{R} \setminus {\pi}$$—which is now done explicitly.

• Added the graphs of sine and cosine, and computed their values in some frequently used points.

• Added an appendix about triangles, and therein computed their values for some more points.

• Added an appendix about generic properties of periodic functions.

• Preprint update, December 14, 2023. Updated the Introduction, to reflect that discussing and defining $$\pi$$ is only done in Section 5.

• Preprint update, December 11, 2023. The extension of sine cosine from $$[0, \pi]$$ to $$\mathbb{R}$$ is now done first, in one go. And only after that is taken care of, is continuity of both functions proved, for all of $$\mathbb{R}$$. In this manner it becomes simpler to follow the reasoning.

• Preprint update, December 10, 2023. Explicitly proved the continuity of sine and cosine, both going from $$[0, \pi]$$ to $$[\pi, 2\pi]$$, and then from $$[0, 2\pi]$$ to $$\mathbb{R}$$.

• Preprint update, December 9, 2023. Corrected a few typos, and clarified the derivation of sine and cosine’s Taylor expansions.

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

• 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.