Trigonometric reduction formulas

[Jan 11]

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.

           

[parisse]

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.

[Jan 11]

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.

[Nigel (UK)]

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)

[parisse]

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.

[Nigel (UK)]

(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)

[parisse]

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...

[parisse]

Update: I have found an easy way to handle tan(pi/2-x) and variants, it should be available in a future firmware update.

[Jan 11]

Parisse, thank you very much.

[CyberAngel]

(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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