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Date: Thu, 18 May 2000 09:57:14 +0300 (EET DST)
Yritän hieman aktivoida palstaa ja näyttää esimerkeillä, minkälaista
keskustelua erilaisilla listoilla käydään. Edellisenkaltaisten
huonoa "netikettiä" edustavien heittojen välttämiseksi ehdotan
myös muulle Solmun toimituskunnalle, että anonymiteettimahdollisuus
poistettaisiin.
Tässä on trigonometrisia funktioita koskevaa pohdintaa. Samalla
saadaan tietokoneilla ja laskimilla tapahtuvan laskennan
lähihistoriaa sekä matemaattisten menetelmien ja laskennan välistä
suhdetta.
Otteet ovat listalta [email protected]
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Shlok Smari Datye wrote:
Hi
What really is Sin, Tan and Cos? I have learnt about it in school and have
calculated degrees in triangles and that kind of stuff. But how does it
work? I meant, I just push some buttons on the calculator without even
knowing how it works! Can anyone explain to me how it works? Is it
possible to find it without the calculator?
Thanks,
Shlok
----------------------------------------------------------------------
Dear Sean.
Thank you very much for your answer. I understood what you explained to me,
and now I understand better how all this works. But one more thing, you
wrote:
"It's from this angle that we can say what your coordinates are -- say you
are at an angle of 30 degrees -- then you'll be at (sqrt(3)/2, 1/2), or
approx (.866,.5). A simple way of finding out these numbers -- your x
coordinate is the Cosine of the angle, and the y coordinate is the Sine of
the angle. In fact, that's how those functions are defined. "
I understand how they are defined, but what if the angle is for example 67
degrees? How did people calculate that? Did they just use a ruler? (I'm
doing all this in hurry and I can't think of anything right now). I heard
that in the old days, they made huge books with all the values, which are
now built inside calculators!
Thanks,
Shlok,
-----Original Message-----
From: Sean Strauss [mailto:[email protected]]
Sent: 16. maí 2000 19:23
To: [email protected]
Subject: Re: [DrMath.com] Sin, Tan, Cos ?
Dear Shlok:
The trig functions sin (short for Sine), Cos (short for Cosine), and tan
(short for Tangent) are developed from the characteristics of a unit circle
(a circle of radius 1, with center at the origin).
Imagine that you are a bug traveling on the circle -- say that you start
on the circle at point (1,0), going upwards. As long as you stay on the
circle, your location on the circle can be determined simply by your angle
made with the horizon.
Does this make sense so far? Let me know if this doesn't.
It's from this angle that we can say what your coordinates are -- say you
are at an angle of 30 degrees -- then you'll be at (sqrt(3)/2, 1/2), or
approx (.866,.5). A simple way of finding out these numbers -- your x
coordinate is the Cosine of the angle, and the y coordinate is the Sine of
the angle. In fact, that's how those functions are defined.
Follow me so far? Let me know if you don't.
Finally -- Imagine the slope of the line that extends from the center of
the circle through the point on the circle. (This line is perpendicular to
the line tangent to the circle at that point). This slope is the Sine
divided by the Cosine -- and that defines the tangent.
The entire course of the trig functions is a complex study of circles...
Mailing-List: list [email protected]; contact [email protected]
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Precedence: bulk
List-Unsubscribe:
Date: Wed, 17 May 2000 14:18:33 +0800
Reply-To: [email protected]
Subject: Re: RE: [DrMath.com] Sin, Tan, Cos ?
At 2000/5/17 14:00:00, Shlok Smari Datye wrote:
>Dear Sean.
>
>Thank you very much for your answer. I understood what you explained
>to me, < Snip > ...
>I understand how they are defined, but what if the angle is for example 67
>degrees? How did people calculate that? Did they just use a ruler? (I'm
>doing all this in hurry and I can't think of anything right now). I heard
>that in the old days, they made huge books with all the values, which are
>now built inside calculators!
>
Trigonometric and transcendental functions can be calculated using
an infinite series; you should be able to find examples in various
handbooks and mathematics text. These series are slow to converge in
many cases, and instead functions called `rational approximations'
are used. These are truncated series with carefully adjusted
coefficients, so that the errors resulting from the truncation are
spread over the range for which the series is valid.
There are other means of calculating some functions, using
fractions which have truncated expressions in their numerators and
denominators. These are all very interesting, and leave me shaking
my head in amazement at their ingenuity when I test one using
arguments at small intervals throughout the range.
I once, as a school assignment, had to calculate to ten decimal
places the common logarithms (base 10) of the integers from 2 to 9.
I did this on a mechanical desk calculator (no electricity
involved). This left me with a great respect and appreciation for
the power of electronic computers when they appeared on the scene.
I still have a copy of the proceedings of a meeting at IBM, at the
time when their first computer, other than those found in their
punch card equipment, was all they had. To calculate the
trigonometric functions, they stored a table of values taken at
fixed intervals, and then interpolated. The discussion centered
around the possibility of calculating the function directly from a
series, rather than using interpolation. As you might guess from
this development, a pocket calculator does not store discrete
values, but instead uses a pseudo-continuous function. I say
`pseudo,' because calculators (or computers) do not do infinite
arithmetic (which is of course why we can use rational
approximations, anyway).
A mathematician named Hastings was one of the first to exploit the
rational approximations that I mentioned earlier. He had an unerring
talent for modifying the coefficients of those truncated series, so
he prepared a book, called appropriately `Hastings's Approximations'
and published it, to the delight of everyone in the computer
industry, who bought copies in quantity. Hastings took his
well-gotten gains, quit his job at the think tank where he worked,
and retired to Hawaii. He likely spent little time concerned with
approximations. Those were the days.
There are often occasions when even rational approximations are not
fast enough in an application, so still another interesting form
called `cordics' can be used. Cordics are often used for graphics
displays, which are often an unending succession of trigonometric
calculations.
Had enough?
George Arndt
------------------------------------------------------------------------
Old school buds here:
http://click.egroups.com/1/4057/10/_/22688/_/958601966/
Heikki Apiola
[email protected]
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Date: Fri, 12 May 2000 16:01:52 +0300 (EET DST)
Solmussa on julkaistu monenlaisia tehtäviä, ylioppilakirjoitus-,
TKK:n valintakoe- ja unkarista käännettyjä tehtäviä, monien
hyvinkin erilaisten kirjoitusten yhteydessä esiintyneitä
tehtäviä.
Olisi paikallaan täsmentää hieman tarkemmin, mitkä näistä ovat
erityisen tylsiä ja myös perustella muutamalla sanalla.
Vähintään kannattaisi keksiä muutama lisäadjektiivi (vaikea,
helppo, säälittävä), jotta voisimme asiaan jotenkin suhtautua.
Heikki Apiola
Solmun toimitus
[email protected]
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Date: Thu, 4 May 2000 13:44:27 +0300 (EET DST)
Tehtävät ovat tylsiä ! Ihan tosissaan............................
Nimim. Tehtävistä...
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Viimeksi muutettu: 15. toukokuuta 2000