Re: A help with trigonometric functions, please
- From: "Hero" <Hero.van.Jindelt@xxxxxx>
- Date: 14 Jan 2007 10:01:40 -0800
David C. Ullrich schrieb:
nicegirl wrote:......
.., and then I read about the polar representation
of a complex number. I thought I had understood, but then a guy who
knows a lot of math told me that, actually, when you write z = r( cos
(t) + i sin(t)) and you think about t as an angle, then the
representation is not precise, because t is not an angle. I had made a
comparisson between z and a vector on the plane, and it was clear to
me that the real part r cos(t) was similar of the projection of a
vector on the vertical axis and the imaginary part r sin(t) was like
it's projection on the vertical axis. That was my geometric reasoning,
thinking of t as an angle and of r as a magnitude, a lenght.
But the guy told me this is not precise and, in formal Math, is wrong.
Well, does this mean everything I have studied so far about
trigonometric functions is wrong? Kinda frustrating!
Sharon
No, there's nothing wrong with anything you know, as far
as I can see. There's also an important point sort of
lying behind what the guy was saying, but if he says
no you're wrong, t is _not_ an angle, that's just being
silly.
When you learn trigonometry you learn that an _angle_ has
a sine. From that point of view you need to specify the
units the angle is measured in - I don't know whether
you've ever heard of radians, but a radian is a unit
of angular measure: pi radians is the same as 180 degrees.
_So_ if you're thinking about the sine of angles then
something like "sin(180)" doesn't make sense - you can't
talk about the sine of a number, it's the sine of an
angle (sin(180 degrees) = sin(pi radians), but
sin(pi radians) would be different.)
Now later, especially when you start studying calculus,
there are reasons you need to have a definition of
sin(x), where x is a _number_, not an angle. So
the sine function gets redefined. Let's say the
new function is Sin and the one you know is sin.
Sin(x) is defined for x a number, not an angle,
while sin(t) is defined for t an angle, not a
number. It turns out that
Sin(x) = sin(x radians).
So it's kind of obvious that anything you can do with
Sin you can also do with sin, and vice versa. The
difference is not just silliness, there are good
reasons why x has to be a number in Sin(x).
But they're really just two ways of looking at the
same thing - saying it's _wrong_ to think of t
as an angle in sin(t) is just stupid, you're just
talking about sin instead of Sin.
Note that this notation Sin is just something I made
up for this post - if you say something to someone
about sin versus Sin they won't know what you're
talking about.
He's also right, although he's being kind of stupid
insisting that his point of view is the only
correct one, that the trig functions are best
defined in terms of power series. This is because
actually _defining_ what an angle _is_ is trickier
than you think. It turns out that
Sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
Cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...,
where n! = 1(2)(3)...(n).
An angle is a difference of direction from a common point. It is
measured on a circle around this point. One can divide the circle into
parts of equal length, like 360 degrees, so 1 degree is about the daily
movement of the sun in relation to the fixed stars, when observed from
earth. It can also be measured by the length of the arc between these
directions divided by the radius. With the formulas given by David one
can calcute the length of sin and cos of a given angle in this second
kind of measure.
Is there a difference between a length of a cos and the cos itself?
I would say yes, the part inside the circle of the one line of the
angle projected ( by parallel rays) onto the other line of the angle
gives a "shadow" of length cos ( of the angle between the two lines)
times the length of the radius. You can measure this length, or You can
calculate it.
This guy may be taking the function, by which one calculate the cos,
for the true soul of the cosinus, and the geometry of it as it's body,
but he can only do this, when the numerical output is precisely the
same. When his formula would give different values he had to call his
function by a different name.
And souls without bodies are like ghosts.
With friendly greetings
Hero
.
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- A help with trigonometric functions, please
- From: nicegirl_130
- Re: A help with trigonometric functions, please
- From: David C . Ullrich
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