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Exact Sine and Cosine Values

May 1st, 2008


Dr Ron Knott in the Department of Mathematics at Surrey University is not a name I recognised, but reading his resume, I now realise that I have heard him talk a few times about Mathematics on Radio 4, both on Simon Singh’s 5 Numbers series, and in Melvyn Bragg’s In Our Time podcast.

I was looking for some information about exact values of trigonometric ratios, and came across his most informative site. I was extremely pleasantly surprised to discover that for some values the trigonometric functions give exact solutions in terms of phi, the golden ratio, among other information.

For example, did you know that the cosine of 27 degrees is exactly a half of the square root of (two plus the square root of (two subtract phi)). (One day when I finish writing my own equation display movies, I’ll write that out in a prettier way, Dr Knott’s website tries a little harder than I do). I love that the number 27, which clearly wants to be prime so much it tricks generations of children into thinking it is, the square root of two and the golden ratio are connected inextricably through the circle-based cosine function. Fantastic!

The whole page, indeed the whole of his site in general, is steeped in extremely interesting, and relatively accessible mathematics with Fibonacci numbers, Egyptian Fractions and so on and so forth. It’s mostly a site for KS4 and beyond (14 years old +), with most material for the older students. Some of it is not for the faint-hearted. However, it is a valuable resource for mathematicians of all hues, and well worth a look.