How can I prove my calculator calculates a trigonometric function?

In summary, a scientific calculator will give you the image of the function ##\tan:\mathbb{R}\rightarrow{\mathbb{R}}##, which is ##(-\pi/2, \pi/2)##.
  • #1
mcastillo356
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TL;DR Summary
One book I have states that any scientific calculator works with this function: ##\arctan:\mathbb{R}\rightarrow{[-\pi,\pi]}##. I want to check it
Considering the measure of angles in radians, that are real numbers, the concept of of trigonometric function spreads to all real numbers. Any real number can be considered as an angle of the first circumference and a ##\mathbb{K}## number of circumferences.

We can consider the function ##\tan:\mathbb{R}\rightarrow{\mathbb{R}}##. This function is not bijective, but if we consider, instead of ##\mathbb{R}##, ##[-\pi,\pi]## as the set origin (which is what scientific calculators make), then it is bijective, and it's possible to define the inverse function ##\arctan:\mathbb{R}\rightarrow{[-\pi,\pi]}##

How can I check this function is which it works in my calculator?

Hope to have explained (and translated) well. I'm learning english, as you can clearly see. If it is not admissible, please delete it.

Greetings!
 
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  • #2
Do you mean you want to calculate the inverse tangent of an number on your calculator and check your calculator gives the correct answer?
 
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  • #3
I knew something was missing. No. I want to know which output does it calculate my device for the function ##\arctan##. Is it really ##[-\pi,\pi]##?
 
  • #4
mcastillo356 said:
I knew something was missing. No. I want to know which output does it calculate my device for the function ##\arctan##. Is it really ##[-\pi,\pi]##?
I doubt it. What answers does it give for ## \arctan 10\ 000 ## and ## \arctan -10\ 000 ## ?
 
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  • #5
mcastillo356 said:
I knew something was missing. No. I want to know which output does it calculate my device for the function ##\arctan##. Is it really ##[-\pi,\pi]##?
As they used to say in Scooby Doo: there's only one way to find out.

Although it should ##(-\pi/2, \pi/2)##.

Excel has no problem calculating ##\arctan##. Not sure why it should.
 
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  • #6
THis is an interesting and tough question to answer. We all trust our calculators to work correctly and accurately. Of course, there are some obscure failures that folks run into such as the Pentium bug or the mystery of ...

https://en.wikipedia.org/wiki/Pentium_FDIV_bug

https://en.wikipedia.org/wiki/Pentium_F00F_bug

or in Windows calculator when doing sqrt operations:

https://www.windowslatest.com/2018/...culator-bug-in-latest-windows-10-build-17639/

or this great video from Matt Parker on the ##11^6 / 13 = (156158413/3600) \pi## oddity on some calculators:



Lastly, for the historical record it was common practice for publishers of navigation tables to insert known mistakes to catch plagiarized tables from other publishers in England. However what was more likely was simple human error in calculating the values. It was one of the reasons why Babbage was commissioned to design and build his Difference engines.

https://www.npr.org/templates/story/story.php?storyId=121206408Some other examples of fictitious entries to catch plagiarists:

https://en.wikipedia.org/wiki/Fictitious_entry

It may well happen that some calculator company might include an obscure bug to catch software theft. This idea was floated in the Matt Parker video too.
 
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  • #7
Long live Charles Babbage.
##\arctan{10000}=1,570696327<\pi<2\pi##
##\arctan{-10000}=-1,570696327<-\pi>-2\pi##
Is this right? And if it's right, what does it mean?##[-\pi,\pi]## is the image of the function ##\arctan## for my calculator?
 
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  • #8
mcastillo356 said:
Long live Charles Babbage.
##\arctan{10000}=1,570696327<\pi<2\pi##
##\arctan{-10000}=-1,570696327<-\pi>-2\pi##
Is this right? And if it's right, what does it mean?##[-\pi,\pi]## is the image of the function ##\arctan## for my calculator?
No, the image of ##\arctan## is ##(-\pi/2, \pi/2)##.

It can't be anything else.
 
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  • #10
Yes, Perok, I'm rambling (I mean just wondering aimless). Yes, the image of ##\arctan## is ##(-\pi/2,\pi/2)##.
So I'd like to encourage you to wonder too: ##\arctan:\mathbb{R}\rightarrow{[-\pi,\pi]}##, is it the way scientific calculators work?. I mean, everyone. I have the feeling I'm asking you to do my homework. But, for example, ##\tan\pi/2=\mbox{Math ERROR}##, but is different, is ##+\infty##
 
  • #11
mcastillo356 said:
So I'd like to encourage you to wonder too: ##\arctan:\mathbb{R}\rightarrow{[-\pi,\pi]}##, is it the way scientific calculators work?. I mean, everyone. I have the feeling I'm asking you to do my homework. But, for example, ##\tan\pi/2=\mbox{Math ERROR}##, but is different, is ##+\infty##
What comes out of a scientific calculator depends on how the calculor has been programmed/designed. But, I'd be surprised if any calculator gives you anything outside the range of ##(-\pi/2,\pi/2)##.

Also ##\tan\pi/2## is undefined, so Math ERROR is correct.
 
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  • #12
When you write the interval as ##( -\pi/2, \pi/2)## it means the end points are not included in the interval also known as an open interval.

The calculator returning MATH ERROR is appropriate and correct.
 
  • #13
I will ask for advice at the store, they will tell me how to manage.
Thanks, @PeroK, @jedishrfu, @pbuk!
 
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  • #14
mcastillo356 said:
I will ask for advice at the store, they will tell me how to manage.
Better yet, look at the documentation for the calculator, which can be found on the manufacturer's website. If you go to the store, I'd bet that the people there would have no idea about the fine points about the calculator.
 
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  • #15
I bought a Casio fx-82MS this morning. The chinese clon is been left aside.
At the guideline, at the point 14. Function Calculations, it mentions ##\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}##, and only adds: "Specify the angle unit before performing calculations".
Well, no problem, but the range of the result, how is it displayed? My question was if this range was ##[-\pi,\pi]##.
Later, it says:
"Pol, Rec: Pol converts rectangular coordinates to polar coordinates, while Rec converts polar coordinates to rectangular coordinates. And adds: "Specify the angle unit before performing calculations. Calculation result ##\theta## is displayed in the range of ##-180º<\theta\le{180º}##
My greyhound doesn't let me continue. I must carry her to the park. See you later
 
  • #16
Greyhound Bus USA always had the motto: Go Greyhound ... and leave the driving to us.

I guess your dog has a similar motto.

greyhound1966b.jpg
 
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  • #17
Her motto is: eat, run, sleep...an infinite loop. Now is sleeping, in a few minutes I will take her to the street to make the last walk before curfew, at 22:00 (in two hours): her name is Pepa :oldlaugh:
20201204_195818_HDR.jpg
 
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  • #18
We have a schipperke with similar habits: eat, run around the house like a greased pig, beg for treats at specific times of the day like clockwork, pester us to take him out as needed and also by 9pm and then pester us to go into his playpen at 10pm for a final treat before sleep.
 
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  • #19
Fiendish, and lovely at the same time.
 
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  • #20
mcastillo356 said:
At the guideline, at the point 14. Function Calculations, it mentions ##\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}##, and only adds: "Specify the angle unit before performing calculations".
The FX82 is capable of using angle units of degrees, radians or gradians. If I remember rightly you can switch between them by pressing SHIFT then MODE.

mcastillo356 said:
Well, no problem, but the range of the result, how is it displayed? My question was if this range was ##[-\pi,\pi]##.
The range for an inverse trig. function will never be ##[-\pi,\pi]##, see answers above.

mcastillo356 said:
"Pol, Rec: Pol converts rectangular coordinates to polar coordinates, while Rec converts polar coordinates to rectangular coordinates. And adds: "Specify the angle unit before performing calculations. Calculation result ##\theta## is displayed in the range of ##-180º<\theta\le{180º}##
And if you choose radians, then polar coordinate angles will be displayed in the range ##[-\pi,\pi]##, but that is not the same as inverse trig. angles.

mcastillo356 said:
My greyhound doesn't let me continue. I must carry her to the park. See you later
We say 'I need to take her to the park'. 'Carry' means that you hold her in your arms, and I don't think she will let you do that :biggrin:
 
  • #21
pbuk said:
The FX82 is capable of using angle units of degrees, radians or gradians. If I remember rightly you can switch between them by pressing SHIFT then MODE.
Yes.
In the USA you work with Texas Instruments calculators? I should have bought one of those, just to communicate with you better

pbuk said:
The range for an inverse trig. function will never be [−π,π]##[-\pi,\pi]##, see answers above.
With this calculator, ##\arctan{10000}=1,570696327## in radians. ##\pi>1,570696327\approx{1,570796327}=\pi/2##
and ##\arctan{-10000}=-1,570696327\approx{-\pi/2}##
So, we have proved the range is ##[-\pi/2,\pi/2]##. It actually it tends to ##[-\pi/2,\pi/2]## when ##\mathbb{R}\rightarrow{\pm{\infty}}##

pbuk said:
And if you choose radians, then polar coordinate angles will be displayed in the range [−π,π], but that is not the same as inverse trig. angles.
This last I haven't took a look. How do Pol and Rec work?

pbuk said:
We say 'I need to take her to the park'. 'Carry' means that you hold her in your arms, and I don't think she will let you do that :biggrin:
Thanks for the remark about this word. I need to improve my english, and you help me.
Pepa wouldn't let me lift and carry her in her whole life. She's a greyhound, a barefooted princess.
Greetings! Sorry for the delay in answering: now it's 6:07 AM
 
  • #22
Sorry, I'm being boring, I would like to know how Pol. and Rec. work. I know, for example, that Pol function calculates the polar coordinates of rectangular coordinates; but I introduce, for example, Pol(1,1), and I should obtain two integers for response: ##(r,\theta )##, but instead the modulus and the angle, I only obtain ##1,414213562=\sqrt{2}=r##; How must I type it to obtain both ##r## and ##\theta##?
 
  • #23
For all I know, the logic gates might all be designed around trigonometry so the jump in logic is in calculating things like 1+1. Q:?)
 
  • #24
I found it!
Example: Convert rectangular ##(1,\sqrt{3})## to polar ##(r,\theta)## (Rad)
##r=2## ##\rightarrow´{}## ##\boxed{Pol(}## ##1## ##,## ##\sqrt{}## ##3## ##)## ##=##
##\theta=1,047197551## ##\rightarrow´{}## ##\boxed{RCL}## ##\mbox{F}##
Press ##\boxed{RCL}## ##\mbox{E}## to display the measure of ##r##, or ##\boxed{RCL}## ##\mbox{F}## for the angle ##\theta##
So...I don't recommend this scientific calculator.
Greetings, UppercaseQ, pbuk, jedishrfu, PeroK!
 
  • #25
mcastillo356 said:
In the USA you work with Texas Instruments calculators? I should have bought one of those, just to communicate with you better
They might do but I'm in the UK where Casio calculators are far more popular as they are in most of Europe I think (although sometimes rebadged under local brands with keys marked in local languages).

mcastillo356 said:
With this calculator, ##\arctan{10000}=1,570696327## in radians. ##\pi>1,570696327\approx{1,570796327}=\pi/2##
and ##\arctan{-10000}=-1,570696327\approx{-\pi/2}##
So, we have proved the range is ##[-\pi/2,\pi/2]##.
Yes indeed; I think it is the same for the other inverse trig. functions. (Edit: strictly speaking the range for arctan is ## (-\frac{\pi}2, \frac{\pi}2) ## because ## \tan \frac{\pi}2 ## is undefined).

mcastillo356 said:
Sorry, I'm being boring, I would like to know how Pol. and Rec. work. I know, for example, that Pol function calculates the polar coordinates of rectangular coordinates; but I introduce, for example, Pol(1,1), and I should obtain two integers for response: ##(r,\theta )##, but instead the modulus and the angle, I only obtain ##1,414213562=\sqrt{2}=r##; How must I type it to obtain both ##r## and ##\theta##?
Not two integers, the modulus is the real number ## \sqrt 2 ##. The display on the FX82 can only display one number at a time so...

mcastillo356 said:
I found it!
Example: Convert rectangular ##(1,\sqrt{3})## to polar ##(r,\theta)## (Rad)
##r=2## ##\rightarrow´{}## ##\boxed{Pol(}## ##1## ##,## ##\sqrt{}## ##3## ##)## ##=##
##\theta=1,047197551## ##\rightarrow´{}## ##\boxed{RCL}## ##\mbox{F}##
Press ##\boxed{RCL}## ##\mbox{E}## to display the measure of ##r##, or ##\boxed{RCL}## ##\mbox{F}## for the angle ##\theta##
... that is how you display the angle.

mcastillo356 said:
So...I don't recommend this scientific calculator.
It's a good calculator and was for some time the one most widely used in UK schools (although this was partly because more advanced calculators were not permitted). Its main disadvantage as you have found is that it can only display one number at a time; calculators with multi-line displays like the FX-85 are easier to work with.
 
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  • #26
pbuk said:
They might do but I'm in the UK where Casio calculators are far more popular as they are in most of Europe I think (although sometimes rebadged under local brands with keys marked in local languages).
At the store I only found Casio.
pbuk said:
Yes indeed; I think it is the same for the other inverse trig. functions. (Edit: strictly speaking the range for arctan is ## (-\frac{\pi}2, \frac{\pi}2) ## because ## \tan \frac{\pi}2 ## is undefined).
Oops...Yes, the range is ## (-\frac{\pi}2, \frac{\pi}2) ##.
pbuk said:
Not two integers, the modulus is the real number ## \sqrt 2 ##. The display on the FX82 can only display one number at a time so...
Thanks! I wrote "integer" without checking the meaning.
pbuk said:
It's a good calculator and was for some time the one most widely used in UK schools (although this was partly because more advanced calculators were not permitted)
What you say encourages me to learn more about characteristics of this calculator.
pbuk said:
Its main disadvantage as you have found is that it can only display one number at a time; calculators with multi-line displays like the FX-85 are easier to work with.
FX-85...Is that the one for universitary students?
Greetings!
 

Related to How can I prove my calculator calculates a trigonometric function?

1. How do I know my calculator is accurate when calculating trigonometric functions?

One way to test the accuracy of your calculator is to compare the results with known values from a table or another calculator. You can also use multiple input values and check if the results are consistent.

2. What is the best way to show that my calculator calculates trigonometric functions correctly?

The best way to prove the accuracy of your calculator is to perform a series of tests using different input values and compare the results with known values. You can also use mathematical proofs to show the validity of the calculations.

3. Can I trust the results shown by my calculator for trigonometric functions?

While calculators are designed to be accurate, there is always a margin of error. It is important to use multiple methods to verify the results and understand the limitations of your calculator.

4. How can I verify the accuracy of my calculator's trigonometric functions without using another calculator?

You can use mathematical identities and properties to verify the results of your calculator. For example, you can use the Pythagorean identity or trigonometric identities to check if the results are consistent.

5. What steps should I take to ensure my calculator is accurately calculating trigonometric functions?

To ensure accuracy, it is important to regularly check and calibrate your calculator. You can also check for software updates and use known values to verify the results. Additionally, understanding the principles behind trigonometric functions can help you identify any errors in the calculations.

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