Trigonometric reduction formulas
03-21-2020, 02:45 PM (This post was last modified: 03-21-2020 04:22 PM by Jan 11.)
Post: #1
 Jan 11 Junior Member Posts: 25 Joined: Mar 2020
Trigonometric reduction formulas
I wonder why HP PRIME does not implement the full list of trigonometric reduction formulas. This applies to the basic trigonometric functions: SIN, COS, TAN, COT for two angle measures (degrees and radians). If we set the calculator in radians, most of the reduction formulas are made. The exception is the TAN function. For example: he will make such a reduction: TAN (π-x) = - TAN (x) or TAN (π +x) = TAN (x). However, it will not do others, e.g. TAN (π / 2-x) or TAN (π / 2 + x) or TAN (3 * π / 2 + x) etc.
If the calculator is set in degrees, it will not perform any trigonometric reduction (see this in the attached screenshots). All competition calculators (TI-Nspire CX-II-T CAS, CASIO CLASSPAD CP-400) have been using these formulas for many years. I think HP PRIME should also have it.

03-22-2020, 01:48 PM
Post: #2
 parisse Senior Member Posts: 1,179 Joined: Dec 2013
RE: Trigonometric reduction formulas
There is almost no support for degree in the CAS, you should always do *exact* trigonometric computations in radians. There are good reasons for that: radians is intrinsic (it's related to the length of the arc) and can be used inside complex exponentials ; derivation, integration, limits and series expansion are too much complicated in degrees.
03-23-2020, 04:53 AM
Post: #3
 Jan 11 Junior Member Posts: 25 Joined: Mar 2020
RE: Trigonometric reduction formulas
If this is the case then complete the missing reduction formulas for the TAN function (in radians). The list of trigonometric reduction formulas should be complete.
03-23-2020, 03:46 PM
Post: #4
 Nigel (UK) Senior Member Posts: 401 Joined: Dec 2013
RE: Trigonometric reduction formulas
If you ask the Prime to simplify $\tan\left({\pi\over 2}-x\right)-{1\over \tan(x)}$ it returns zero, so it knows that the two expressions are equal. However, it won't simplify either one to the other. A possible reason is that it has no way of knowing which form you consider to be simpler.

If I ask the Prime to expand $$\tan(a+b)$$ with the texpand command, it does so. However, if $$a=\pi/2$$ it returns undef. I guess that this is because $$\tan(\pi/2)$$ is indeed undefined, although the Prime does correctly return $\lim_{a\to\pi/2} \Bigl({\rm texpand\,}\left(\tan\left(a-x\right)\right)\Bigr)$ as $$\cos(x)/\sin(x)$$. Maybe the CAS could be given a new rule to allow it to expand $$\tan(a+b)$$ when either $$a$$ or $$b$$ is an odd half-multiple of $$\pi$$?

Nigel (UK)
03-23-2020, 03:47 PM (This post was last modified: 03-23-2020 03:48 PM by parisse.)
Post: #5
 parisse Senior Member Posts: 1,179 Joined: Dec 2013
RE: Trigonometric reduction formulas
cot is not a fundemental trigonometric function inside the CAS, it will always be rewritten with sin and cos.
You can run e.g. sincos(tan(pi/2-x)) if you want to remove a pi/2-multiple phase shift in a tan function.
03-24-2020, 04:30 PM
Post: #6
 Nigel (UK) Senior Member Posts: 401 Joined: Dec 2013
RE: Trigonometric reduction formulas
(03-23-2020 03:47 PM)parisse Wrote:  cot is not a fundemental trigonometric function inside the CAS, it will always be rewritten with sin and cos.
You can run e.g. sincos(tan(pi/2-x)) if you want to remove a pi/2-multiple phase shift in a tan function.

I'm not asking for cot to be given as an answer. It just seems strange that texpand acting on $$\tan(a-b)$$ returns $$(\tan a -\tan b)/(1+\tan a\tan b)$$, but acting on $$\tan((\pi/2)-b)$$ texpand returns "undef". This expression is perfectly well-defined. Returning either $$\cos x/\sin x$$, $$1/\tan x$$, or even the original expression unchanged would be better than "undef".

I appreciate that it is not possible for a CAS - especially on a calculator - to consider every special case in every situation, so I do understand if things are left as they are.

Nigel (UK)
03-24-2020, 07:13 PM
Post: #7
 parisse Senior Member Posts: 1,179 Joined: Dec 2013
RE: Trigonometric reduction formulas
tan(pi/2-b) is well defined, but if you apply the formula that texpands apply, it will involve tan(pi/2) and that's infinity...
Of course it's possible to add a special check for pi/2-b, but that's not so easy if you want to handle all multiples of pi/2, like tan(3*pi/2-b) and of course also tan(b+pi/2), etc.
Unfortunately, I do not have infinite time ressources, I must make choices...
03-25-2020, 07:39 AM (This post was last modified: 03-25-2020 07:40 AM by parisse.)
Post: #8
 parisse Senior Member Posts: 1,179 Joined: Dec 2013
RE: Trigonometric reduction formulas
Update: I have found an easy way to handle tan(pi/2-x) and variants, it should be available in a future firmware update.
03-25-2020, 08:07 AM
Post: #9
 Jan 11 Junior Member Posts: 25 Joined: Mar 2020
RE: Trigonometric reduction formulas
Parisse, thank you very much.
03-25-2020, 04:03 PM (This post was last modified: 03-25-2020 04:07 PM by CyberAngel.)
Post: #10
 CyberAngel Member Posts: 296 Joined: Jul 2018
RE: Trigonometric reduction formulas
(03-25-2020 07:39 AM)parisse Wrote:  Update: I have found an easy way to handle tan(pi/2-x) and variants, it should be available in a future firmware update.

Cyrille? Tim W?
Could we have a new (virus-free) Beta soon (4 the Real calculatrice G1), please!?
This could be a sort of a Crown jewel!
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