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Saturday, September 14, 2013

ARR!.. apparently i still have more trigοnometry to learn.

well, i learned something new today:
It sounds cumbersome now, but doing multiplication by hand requires a lot more operations than addition does. When each operation takes a nontrivial amount of time (and is prone to a nontrivial amount of error), a procedure that lets you convert multiplication into addition is a real time-saver, and it can help increase accuracy.

The secret trig functions, like logarithms, made computations easier. Versine and haversine [1] were used the most often. Near the angle \theta = 0, \cos(\theta) is very close to 1. If you were doing a computation that had 1-\cos(\theta) in it, your computation might be ruined if your cosine table didn’t have enough significant figures. To illustrate, the cosine of 5 degrees is 0.996194698, and the cosine of 1 degree is 0.999847695. The difference \cos(1^o)-\cos(5^o) is 0.003652997. If you had three significant figures in your cosine table, you would only get 1 significant figure of precision in your answer, due to the leading zeroes in the difference. And a table with only three significant figures of precision would not be able to distinguish between 0 degree and 1 degree angles. In many cases, this wouldn’t matter, but it could be a problem if the errors built up over the course of a computation.


~ from "10 Secret Trig Functions Your Math Teachers Never Taught You" @sciam
in other news: it's been more than two weeks into this new job, and i still feel disoriented. often i feel exhausted, too.

on the bright side: i finally found an expensive apartment and signed a lease .. after a month of searching (and simultaneously teaching, for the last 2 1/2 weeks).

[1] these are defined, respectively, as \textrm{versin}(\theta) = 1-\cos(\theta) and \textrm{haversin}(\theta) = \frac{1}{2}\textrm{versin}(\theta). suggestively, "ha" mean half.

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