Trigonometry question

I don't feel this to be a 'question'.....consider it as a doubt instead.

I am not able to understand the need to EXTEND trigonometric definitions.(and so introducing new concepts like unit circle). Please give me some satisfactory reasons.

#Geometry

Note by Anandmay Patel
4 years, 7 months ago

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Comments

Here's an opinion that may irate the crowd in this website: YOU DON'T.

The idea of "sines" and "cosines" (which are actually quite a recent development, within the last 500 years) assumes that you already have a working notion of an "angle", i.e. a separation between two intersecting lines, and "distance", i.e. a separation between two (distinct) points. "Distances" were computed as displacements in ancient times, whereas "angles" (or really their "cosines") can be computed by drawing a circle centred at the intersecting point, so that you have then a set of 3 points for which you get a triangle out of; from there, you compute the ratio of its area (a purely affine quantity in terms of determinants) to the product of their "distances" to get some notion of an "angle". This is a method that predates Ancient Greece; some have opined that the Old Babylonians thought of "angles" in such a way.

The modern idea of "angles" requires a notion of arclength, which is not easy to compute without any limiting processes, and the modern idea of "distances" requires taking "square roots" of square areas. While one can argue that "square roots" are not exact computationally, the larger issue is that in some fields "square roots" do not exist (as an example, consider the finite field of 11 11 elements and try to find a number whose square is 2 2 or 5 5 mod 1111).

The main point of my diatribe is that trigonometry in the 21st century ought to be framed purely in the context of linear algebra and vector geometry. Rather than the idea of "distances", we should treat the separation of two points as a metric by which our geometry operates on; here, we use the powerful tool of inner products. We will also draw on the ancient method of computing "angles", which again can be computed using only linear algebraic tools.

Here's something you can try and think about: think of the geometry on the surface of a (unit) sphere. How do the results of normal, planar trigonometry generalise when you work on a different surface, if it can be done at all? Are there any subtle differences? How about if we started stripping usual Euclidean axioms?

A Former Brilliant Member - 2 years, 10 months ago
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