If you like that kind of surprise you might wish to revisit cos^2 x + sin^2 x = 1. Try it with x a complex number. Why on earth does that work? It is not as tho

If you like that kind of surprise you might wish to revisit cos^2 x + sin^2 x = 1. Try it with x a complex number. Why on earth does that work? It is not as though a complex number makes sense as an angle. More at http://lydianrain.wordpress.com/2009/01/06/summoning-cthulhu-sin2-cos2-with-power-series/ http://lydianrain.wordpress.com/2009/01/06/summoning-cthulhu... You could also look at Chebyshev polynomials T_0(x) = 1 T_1(x) = x T_2(x) = 2x^2 - 1 T_3(x) = 4x^3 -3x It is easy enough to find recurrance relations and plots on the web. Notice that all the graphs go from -1 to 1 and illustrate that the polynomials are strangely well behaved in this range, oscillating up and down between -1 and 1. Why is that? T_n(x) = cos (n arccos x) There is this nice trignometric formula, but since cosine is always between -1 and 1 it makes no sense to try using its inverse function, arccos, outside this range. On the other hand the polynomials have no implied restriction on x, so if you have a calculator, such as a Common Lisp REPL, with full support for complex numbers you can use the trignometric formula outside the -1 to 1 range and observe it agreeing with the polynomials.