[Python-ideas] Fwd: Trigonometry in degrees

Steven D'Aprano steve at pearwood.info
Mon Jun 11 01:48:31 EDT 2018

```On Sun, Jun 10, 2018 at 10:01:09PM -0700, Chris Barker via Python-ideas wrote:

> In regard to the "special values", and exact results -- a good math lib
> should return results that are "exact" in all but maybe the last digit
> stored. So you could check inputs and outputs with, e.g. math.isclose() to
> give people the "exact" results. -- and keep it all in floating point.

I wish Uncle Timmy or Mark Dickinson were around to give a definite
answer, but in their absence I'll have a go. I'm reasonably sure
that's wrong.

The problem with trig functions is that they suffer from "the
table maker's dilemma", so it is very hard to guarantee a correctly
rounded result without going to ludicrous extremes:

http://perso.ens-lyon.fr/jean-michel.muller/Intro-to-TMD.htm

So I think that there's no guarantee given for trancendental functions
like sine, cosine etc.

But even if they were, using isclose() is the wrong solution. Suppose
sin(x) returns some number y, such that isclose(y, 0.0) say. You have no
way of knowing that y is an inaccurate result that ought to be zero, or
whether the answer should be non-zero and y is correct. You cannot
assume that "y is close to zero, therefore it ought to be zero".

It's not just zero, the same applies for any value. That's just moving
rounding errors from one input to a slightly different input.

# current situation
sine of x returns y, but the mathematical exact result is exactly z

# suggested "fix"
sine of x ± a tiny bit returns exactly z, but ought to return y

Guessing what sin or cos "ought to" return based on either the inexact
input or inexact output is not a good approach.

Remember, because π is irrational, we cannot actually call sin or cos on
any rational multiple of π. We can only operate on multiples of pi,
which is *close to* but not the same as π. That's why it is okay that
tan(pi/2) returns a huge number instead of infinity or NAN. That's
because the input is every so slightly smaller than π/2. That's exactly
the behaviour you want when x is ever so slightly smaller than π/2.

--
Steve
```