For some reason I developed an urge to do some arithmetic calculations. There is supposedly a narrative from a science fiction short story where planetary explorers are returning to earth and all their computers go down. An historian that they had brought along teaches the crew and passengers the old style by-hand arithmetic algorithms and they get home safely. So, since they were doing navigational computing, I decided to calculate the sine of an angle. I chose the angle, .11 radians, because multiplying by eleven is so much fun. My rule was absolutely no use of a calculator. However I did use one to calculate sin(.11) before I started so I could compare what I was getting with the correct number. So much for emulating the science fiction story. I checked each calculation by casting out nines and casting out eleven’s. It was very hard to resist grabbing the calculator when things didn’t check. It also took me a little time to figure out how to use casting out to check division.
My original intention was to calculate the sin(.11) in the same way as the calculator. My research found that calculators use some sort of interpolation method like the CORDIC algorithm or Chebyshev polynomials. I wasn’t about to calculate a bunch of other points by hand so I just used the Taylor series approximations. Since .11 is so small the calculation converged rapidly for the precision I wanted – that of a TI-83 calculator.
Here are the calculations.