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I do not have access to a TI Inspire CAS, and was wondering if someone can tell me the result for the SIN of exactly 3.141592654 radians on it. Thanks
PS, I know what the correct answer is to as many digits as I care to get, (via Mathematica), but curious how accurate the TI is compared to some HPs that get it correct to 12 significant digits ...TIs in the past have not been as accurate as certain HPs for Trig functions...
My TI-nspire cx CAS, given the following in RAD mode:

sin(3.141592654)

gives:

-4.102e-10
That's pretty accurate.

For values of x around $$\pi$$, $$sin(x) \approx \pi-x$$

The value of $$\pi$$ is closer to 3.14159265358979, and $$3.14159265358979-3.141592654=-4.1021\times10^{-10}$$ so I'd say this machine is right on the money.

The inaccuracy here isn't in the calculator, it's in the assumed value of $$\pi$$.
Hallo Zeno33,

on a TI nspire CX (not the CAS model) I got

sin(3.141592654)=-4.102E-10

I hope that helps,

Thomas
Hmm - you're only getting four digits of result. Normally we'd say that's not very accurate.

For this kind of calculation the calculator does need to do more work than usual - it helps to have an internal value for pi which is twice the number of digits displayed - but from a user perspective, how it does it isn't the same question as how accurate it is. Because the input value is close to pi, there's a possibility of loss of precision, which is what we see here.

To be fair to TI, most calculators don't do much better and may even do worse.
(01-29-2019 07:58 AM)EdS2 Wrote: [ -> ]Hmm - you're only getting four digits of result. Normally we'd say that's not very accurate.

Unless the display is set to FIX 4 (or its equivalent in TI-land).
(01-29-2019 02:41 AM)zeno333 Wrote: [ -> ]PS, I know what the correct answer is to as many digits as I care to get, (via Mathematica), but curious how accurate the TI is compared to some HPs that get it correct to 12 significant digits ...TIs in the past have not been as accurate as certain HPs for Trig functions...

Curious, what answer do you expect? Precisely 0? In that case the sin function would have edge cases hardcoded is the most likely case.
FWIW, my TI-30X Pro MathPrint displays the answer -4.102067615e-10, with an internal result of -4.102067615374. That seems to be pretty accurate. (For comparison, HP-42S shows -4.10206761537e-10, and DM42/Free42 shows -4.102067615373566167089928953969909e-10.)
(01-29-2019 11:56 AM)pier4r Wrote: [ -> ]
(01-29-2019 02:41 AM)zeno333 Wrote: [ -> ]PS, I know what the correct answer is to as many digits as I care to get, (via Mathematica), but curious how accurate the TI is compared to some HPs that get it correct to 12 significant digits ...TIs in the past have not been as accurate as certain HPs for Trig functions...

Curious, what answer do you expect? Precisely 0? In that case the sin function would have edge cases hardcoded in the most likely case.

Yeah, I'd rather get the real answer rather than have the calculator try to fudge by showing 0.
(01-29-2019 07:58 AM)EdS2 Wrote: [ -> ]... it helps to have an internal value for pi which is twice the number of digits

We need even more digits of pi to handle bigger argument.
Example, assumed all calculation in 24 digits precision :

sin(411557987)
= sin(411557987 mod (2 * pi))
~ sin(411557987 mod (2 * 3.14159265358979323846264))
= sin(2.53671649518336E-9)
= 2.5367164951833599972794E-9

Rounded back to 12 digits, sin(X) = 2.53671649518E-9.
24 digits math only get 7 accurate digits !

Using spigot calculator:
> spigot --printf=%.12g sin(411557987)
2.53671605196e-09

see FSIN "bug": Intel underestimates error bounds by 1.3 quintillion
AFAICT, -4.102e-10 is the most precise answer for sin(3.141592654) on the TI Nspire CX CAS running OS 4.5.0.1180. I've been through various display settings and have been unable to extract any more digits of precision.
(01-29-2019 06:07 AM)grsbanks Wrote: [ -> ]That's pretty accurate.

For values of x around $$\pi$$, $$sin(x) \approx \pi-x$$

The value of $$\pi$$ is closer to 3.14159265358979, and $$3.14159265358979-3.141592654=-4.1021\times10^{-10}$$ so I'd say this machine is right on the money.

The inaccuracy here isn't in the calculator, it's in the assumed value of $$\pi$$.

I was not assuming anything...I was asking for the SIN of a specific input, and the HPs give a much higher accuracy than the TIZs...The HP 28, 42S, 48 series 50G and prime etc give the answer correct to 12 digits, the TI is not even close.
(01-29-2019 11:56 AM)pier4r Wrote: [ -> ]
(01-29-2019 02:41 AM)zeno333 Wrote: [ -> ]PS, I know what the correct answer is to as many digits as I care to get, (via Mathematica), but curious how accurate the TI is compared to some HPs that get it correct to 12 significant digits ...TIs in the past have not been as accurate as certain HPs for Trig functions...

Curious, what answer do you expect? Precisely 0? In that case the sin function would have edge cases hardcoded in the most likely case.

The correct answer to 12 digits is --4.10206761537 E-10...This problem is described in detail in the optional HP 15C Advanced Functions Handbook...
(01-30-2019 02:32 AM)zeno333 Wrote: [ -> ]I was not assuming anything...I was asking for the SIN of a specific input, and the HPs give a much higher accuracy than the TIZs...The HP 28, 42S, 48 series 50G and prime etc give the answer correct to 12 digits, the TI is not even close.

I'd say that depends on the particular model. The HP-41 series, the 67, 15C and – I suppose – other 10-digit devices return 4,10 E–10. Which is what you get with a pi constant rounded to the internal 13 digits of these calculators.

Dieter
For those still unsure, the key here is that 3.1415926534 is not pi, it is an approximation to pi. So, an exact calculation of sin(x) using this number should not be zero - it should be approximately zero. (Edit: whereas sin(180) in degree mode should be exactly zero.)

Interestingly enough, HP themselves said the same thing, in 1976:

Quote:There is a point where the new algorithms appear at first glance to be less accurate. If we calculate tan π in radians, the answer is 4.10 E -10 on the HP-91 whereas the HP-55 yields 0. This is misleading, for π on the keyboard should really be labeled π₁₀, showing that it is ten digits of π. Thus
• π₁₀ = 3.141592654
To ten digits, tan π₁₀ = 4.120676150 E -10. One can see that the HP-91 gives the better answer to this problem.

The reason that the HP-91 does not give the correct answer is that prescaling is done with a finite-length π. To give the correct answer to this problem, a π of length 20 would have to be used. It can be shown that to produce the correct answer to all trigonometric problems in radian measure, it takes a π of length 120. The HP-91 actually uses 13 digits of π.
See
The New Accuracy: Making 2³ = 8, by Dennis. W. Harms
in HP Journal, November 1976 page 17.
Hello!

If you are using the sin(pi) on a TI-nspire you get zero.

But I think, the value of 3.1415926534 for pi is not a good choice.
For calculation in real world physics normally you don't need so many significant digits, so I think using pi directly is ok.

And I have tried several calcs:
DM-42: 1.897932384626433832783634455164435e-10
DM-15: 5.9e-10
fx-991DE PLUS: 1.898e-10
HP-300s+: 1.898e-10
numworks: 1.8997932e-10
HP-35s: 1.89793238e-10
2DS: 0
I have input the number from the last post 3.1415926534 (and not 3.141592654) with all digit if possible.
As you can see one digit more, you get a result half as big as the other (-4.102069e-10 by numworks).
So I think rounding to 0 is fine for real world problems.
That's the reason HP choice FIX 4 as default for many calcs.
Bernd
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