6/1/2023 0 Comments Mathematica pi![]() However, there is a section in Gauss's Disquisitiones Arithmeticae (sorry, I couldn't find a free English translation) that relates to a number-theoretic view of the roots of unity (and thus, trigonometric functions). Now, "Gauss's algorithm" is not terribly informative, considering that Gauss was a rather prodigious producer of mathematical results. The kicker is that ToRadicals and Developer`TrigToRadicals return different-looking (but numerically equivalent) results, which means the internal algorithms have been changed somewhat. Using Developer`TrigToRadicals throws an error message saying that it is obsolete, and that one should now use ToRadicals instead. Now, however, FunctionExpand no longer seems to perform this conversion. This hinged on the function Developer`TrigToRadicals, and in fact one could use this directly instead of FunctionExpand. Indeed, in the old versions of Mathematica, FunctionExpand certainly was able to convert trigonometric values at rational multiples of $\pi$ to the corresponding radical values (possibly involving complex numbers). I think writing this answer is as good a time as any to bring it up.įirstly, there is this tantalizing line from the internal implementation notes:įunctionExpand uses an extension of Gauss's algorithm to expand trigonometric functions with arguments that are rational multiples of $\pi$. ![]() In the course of answering this question, I ran into a little bit of weirdness that doesn't square with my experience with previous versions of Mathematica. ![]()
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