Quoting Dan Asimov <dasimov@earthlink.net>:
Mike wrote:
<< Is there a geometric way to understand the Taylor series for sin and cos? The closest I've been able to find is a combinatorical explanation (below), but it doesn't seem to help much.
I'd love to know such a way to understand their Taylor series.
I recently gave a talk to a bunch of smart youngsters about complex numbers, and was unable to find a truly graceful way to explain why
exp(ix) = cos(x) + i sin(x),
(without deriving their Taylor series).
So if anyone knows a way to see this, I'd love to know it.
But of course that would require giving exp a meaning on the imaginaries.
I don't know about complex categories (?) but finding a way to get Euler's formula has interested me for a long, long time. From calculus, the series for sine and cosine can be merged, but what if you don't know about calculus and infinite series yet. An approach I have seen is to define cis (cosine plus i times sine) and use trigonometric identities to manipulate it; it is an ersatz exponential and leaves any connection with an actual exponential unanswered. It meshes nicely with using polar coordinates in the complex plane, but doesn't have the radial factor. Anyway, I finally understood things to my satisfaction by using 2x2 real matrices to represent multiplication in the complex plane and using tghe definition of the exponential as the limit of the product (1 + (small angle) x matrix representing complex number)). The coefficients of the series come from the binomial formuls, taking into account that the matrix product must be ordered. Matching that with cis gives the desired result, with the taylor's series as a byproduct, not as an initial hypothesis. The series convergrs to a rotation by neatly fencing the rotated vector in by a sequence of contracting boxes. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos