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 Definition In a right triangle, the ratio of the length of the side opposite an acute angle (less than 90 degrees) and the length of the hypotenuse. [d] [e]
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Geometry v. calculus

Dmitrii's calculus/analysis introduction of sine and cosine is interesting and definitely worth giving in this article. Yet, it seems to me that the standard geometric approach should be given as well, because CZ is an encyclopedia, where one expects to find world's standard knowledge presented as traditionally as possible. Therefore, unless I hear that there is strong disagreement, I will write a first introductory section with a few drawings illustrating the geometric definition and graphs of sine and cosine. Also the formula's for sin(α+β), and so on, are needed.

The definition of Pi (π), at the end of the article, is also interesting (and new to me), but would it not be better to move this to the article on Pi as one of the ways that Pi may be defined?--Paul Wormer 08:27, 1 November 2008 (UTC)


(This continues a discussion on my talk page, where I was wondering whether we should introduce vectors). I did a quick and dirty rewrite of the definition using Cartesian coordinates without using vectors:

=== Definition using the unit circle ===

We recall that a point P in a 2-dimensional plane can be represented by two Cartesian coordinates, which are real numbers. These numbers are the projections of P on the x- and the y-axis, respectively. The point P lies on the circle with centre at the origin and radius one if its coordinates satisfy

We call this circle the unit circle.

Consider Fig. 1 again. Assume now that A lies at the origin, that AB lies on the x-axis and that the length b equals 1. Then, the point C lies on the unit circle. Clearly, cos α = c is the x-coordinate of the point C. Similarly sin α = a is the y-coordinate of C. This observation leads to the following definition, which is a generalization of the earlier one.

Definition: If C is a point on the unit circle and AC makes an angle α with the x-axis, measured counter-clockwise, then cos α is the x-coordinate of C and sin α is the y-coordinate of C.

I think that's a bit easier to understand. However, as Paul said, the article uses unit vectors in the proof of the sum formula, so I'll also have to think about how to do that part without vectors. Unfortunately, I ran out of time. So to be continued. Or, if you think the vector approach is better (and I agree with Paul that the difference is minimal and mostly semantics), please say so and I won't waste my time. -- Jitse Niesen 15:11, 13 November 2008 (UTC)

When we decide to get rid of vectors, I can easily adapt the figures and replace arrow points by small dots indicating points in the plane (or something similar).--Paul Wormer 08:44, 14 November 2008 (UTC)