The Student Room Group
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>

Trigonometric` functions definition

Hi. I was wondering how exactly people/ calculators know exactly what the trig functions look like. I mean- how would calculators know exactly how to work out trig functions. I know it's obvious from a wiki animation of a unit circle and the graph (http://en.wikipedia.org/wiki/File:Circle_cos_sin.gif) why it looks like this- but are there any sorts of advanced definitions of the functions or, formulae that a calculator uses for example to work it out?
I've read something saying that there are series definitions of the trig functions like a Taylor series or something- would somebody please elaborate on this? (I don't know whether this is true)
Thanks! :smile:
Original post by MathMeister

I've read something saying that there are series definitions of the trig functions like a Taylor series or something- would somebody please elaborate on this? (I don't know whether this is true)
Thanks! :smile:



http://en.wikipedia.org/wiki/Taylor_series
Original post by TenOfThem

Ahh so a calculator uses this. It is defined as a series...
Original post by MathMeister
Ahh so a calculator uses this. It is defined as a series...


no, I believe not

http://www.homeschoolmath.net/teaching/sine_calculator.php


btw - I googled those links !
Original post by TenOfThem
no, I believe not

http://www.homeschoolmath.net/teaching/sine_calculator.php
btw - I googled those links !

Thanks-yh i suppose I could've googled it myself lol
Original post by MathMeister
Hi. I was wondering how exactly people/ calculators know exactly what the trig functions look like. I mean- how would calculators know exactly how to work out trig functions. I know it's obvious from a wiki animation of a unit circle and the graph (http://en.wikipedia.org/wiki/File:Circle_cos_sin.gif) why it looks like this- but are there any sorts of advanced definitions of the functions or, formulae that a calculator uses for example to work it out?
I've read something saying that there are series definitions of the trig functions like a Taylor series or something- would somebody please elaborate on this? (I don't know whether this is true)
Thanks! :smile:


They (handheld calculators) use something called the CORDIC algorithm (look it up). Although the algorithm actually requires tables of numbers to work, specifically arctan(2−k);k=0,1,2,...,narctan(2^{-k}); k=0,1,2,...,n where n is related to the number of iterations in the CORDIC algorithm (essentially how accurate you want the answer). These tables can be produced by taylor series and are actually stored in the calculator so it does not calculate them.
Original post by tombayes
...

Integration can sometimes take quite some time on my 991ES plus. Do you reckon the calc uses some sort of estimate for these as-well based on some formula like the Simpsons one or other better ones say?
Original post by MathMeister
Integration can sometimes take quite some time on my 991ES plus. Do you reckon the calc uses some sort of estimate for these as-well based on some formula like the Simpsons one or other better ones say?


The this


x
Original post by MathMeister
Integration can sometimes take quite some time on my 991ES plus. Do you reckon the calc uses some sort of estimate for these as-well based on some formula like the Simpsons one or other better ones say?


Not sure. But they would probably not use the simpson' s rule - if you have done any programming you will see it is not difficult to write a program to integrate numerically based upon simpson's rule but getting it efficient is very hard (you end up with many cases were it does not work and have to look at them separately).

So low power devices would probably use a different method like the Gauss-Kronrod Quadrature method or some others like Clenshaw-Curtis or something else. But this is not perfect e.g. ∫010cos⁡(πx)dx\int_0^{10} \cos (\pi x) dx should be 0
Original post by TenOfThem
no, I believe not
btw - I googled those links ! ikrbut tsr is way better than googlelol

So for example sin(x) is just notation for the what is actually an approximation?
How did they find the graph in the first place? Did they draw it out measuring the angles/ lengths or is there a sort of rate of change of angle lets say with respect to the length? And from there they graphed it?- but then they'd need some sort of formula and knowledge of how the sin(x) graph it's now known behaves.- otherwise they'd of just drawn an incorrect wave curve not looking exactly like the proper one...
plz? anybodeh?
Original post by MathMeister
So for example sin(x) is just notation for the what is actually an approximation?
How did they find the graph in the first place? Did they draw it out measuring the angles/ lengths or is there a sort of rate of change of angle lets say with respect to the length? And from there they graphed it?- but then they'd need some sort of formula and knowledge of how the sin(x) graph it's now known behaves.- otherwise they'd of just drawn an incorrect wave curve not looking exactly like the proper one...
plz? anybodeh?


http://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you