Euler Identity in Home

[jebem]

Thanks, Joe and Peacecalc for your explanations. It makes all the sense.

So, sin(Pi) can be displayed with a result <> Zero because it depends on two factors:
- Internal accuracy (how many digits it can handle for the calculations);
- How close to Zero has a result to be, in order to be displayed as Zero.

Apparently HP has followed a different route for their more recent top models, when comparing the behavior of what CASIO and TI are doing.

If I understand correctly, HP is not adjusting the results to Zero, no matter how small is the value (?), where CASIO and TI are doing that and also they use at least 14 digits of accuracy compared to 12 digits for HP (?).

Please have a look on my tests (on real calculators) below for details:

Calculator, SIN(Pi), SIN(Pi/2), COS(Pi), Remark#
-----------------------------------------------------
HP-65, 0, 1, -1
HP-25, 0, 1, -1
TI-57, 0, 1, -1
Texas TI-51-III, 0, 1, -1
Casio FX-39, 0, 1, -1
Casio FX-5000F, 0, 1, -1
HP-300S+, 0, 1, -1

Texas TI-85, 0, 1, -1, Remark1
Texas TI-89, 0, 1, -1, Remark2

Casio AFX 2.0+, 0, 1, -1, Remark3

HP-48G+, -2.06761537357E-13, 1, -1, Remark4
HP-35S, -2.06761537357E-13, 1, -1
HP-49G+, -2.06761537357E-13, 1, -1, Remark5
HP-48GII, -2.06761537357E-13, 1, -1, Remark5
HP-PRIME, -2.06761537357E-13, 1, -1, Remark6

Remarks:
(1) Texas TI-85: This is not a CAS capable calculator.
"SIN Pi" = 0 all the time, even when we convert the "Pi" value to a number by doing "Pi ENTER", and then "SIN 2nd ANS";
Despite the displayed "Pi" value only having 12 digits, internally it has 14 digits of accuracy;
To prove it, one needs to manually type 12 digits for "Pi" and do a "SIN 3.14159265359" to obtain this time a result of "-2E-13", very close to the recent HP top models result;
As stated in the user guide: Internally "Pi" is a constant = "3.1415926535898" with 14 digits of accuracy, but the calculator displays only up to 12 digits.

(2) Texas TI-89: It computes "sin(Pi)" = 0 all the time, even when we convert the "Pi" to a number by doing "Pi ENTER", and then "sin(2nd ANS)", sporting the 14 digits of accuracy internally;
Contrary to the older TI-85, the TI-89 can also display the "Pi" value with a full 14 digits;
Now, doing a "sin(3.1415926535898)" with 14 digits by manually typing this value for "Pi", we get a result = 0, so there is no symbolic calculation going on here.
And if we just use 12 digits for "Pi" and do a "sin(3.14159265359)" we obtain this time a result of "-2E-13" as with the TI-85, again very close to the recent HP top models result;

3) Casio Algebra FX 2.0 plus: It computes "sin Pi" = 0 in computing mode (non-CAS), even when we convert the "Pi" to a number by doing "Pi ENTER", and then "sin SHIFT Ans";
Also, doing a "sin 3.1415926535898" by manually typing 14 digits for "Pi", we get a result = 0, the same result as obtained for Texas TI-89, so the internal accuracy is at least 14 digits, and there is no symbolic calculation going on here as well.
And if we just use 12 digits for "Pi" and do a "sin 3.14159265359" we obtain this time a result of "-2E-13", again very close to the recent HP top models result.

(4) HP-48G+: Needs to convert "Pi" to a numeric value (with "-> Num"), otherwise it gives "SIN(Pi)" = 0;

(5) HP-49G+, HP-48GII, HP-50G: Needs "CAS Numeric" selected, otherwise it gives "SIN(Pi)" = 0;
Now, if we type "SIN(3.1415926535898) we get a result of 9.79323846264E-12 and the inputted command is just truncated in history to have only 12 digits: "SIN(3.14159265358)".

(6) HP-PRIME: Needs Home mode selected, otherwise in CAS mode it gives "sin(pi)" = 0;
Even if we type "SIN(3.1415926535898) we still get a result of -2.06761537357E-13, as the inputted command is rounded in history to have just 12 digits: "SIN(3.14159265359)".
So it seems that the PRIME shows a better behavior than the HP-50G in this regard.