35s...Cosine of large numbers Message #3 Posted by Hal Bitton in Boise on 14 Jan 2009, 1:26 a.m., in response to message #1 by Chuck
Interesting...
My first instinct would be to say that as the coefficients of 2pi get bigger, progressively more of the calculators available numerical precision is taken up by the integer portion of the resultant large number. And of course the fractional part of pi is what defines that irrational constant...the more decimal places, the better. As you truncate decimal places, you loose accuracy, etc. What's interesting here is that if I reduce the domain manually to <2pi of any of the three figures you cited (by dividing by 2pi and multiplying the remainder (or fractional part) by 2pi). I always get zero, and hence a cosine of 1. I tried this in exact mode, carrying pi symbolically throughout, and also in approximate mode, carrying pi numerically. In fact, this proved to be the case for 2pi multiplied by any exponent of ten. I think this is simply because the fractional part of any number bigger than about 15 digits is always zero (as for as the calculator is concerned). This works out well for any even multiple of 2pi, but would be disastrous for anything else, (such as 2pi x10^25 + pi/4), which explains why the calculator obviously doesn't use this method for normalization of large arguments for trig functions.
For what it's worth, I did note that accuracy for trig functions in degree mode is dead on for 360*10^x for x=0497 :)
Best regards, Hal
