Re: A help with trigonometric functions, please

On 14 Jan 2007 07:17:59 -0800, nicegirl_130@xxxxxxxxx wrote:

Hi, I'm a 15 yo girl and I like to study math more than required by my
school. Some days ago I asked a help with complex numbers, the answers
were quite interesting, 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.
Though I didn't understand what he meant, I could see this was about
the very definition of the sine and cosine functions The definitions I
have of these functions are those based on right triangles or, maybe
more precise, on the unit circle. But he told me those definitions are
not very accurate, because their right definitions of sin and cosin are
based on something he called power series. The guy didnt want to waste
his time with someone my age and didnt go into details, but I think a
power series is something like a polynomial of infinite degree with
infinitely many coefficients. I have studied geometric progressions and
I know that if |x| <1 then 1 + x + x^2....+ x^n...... = 1/(1-x), that
is, when n goes to infinity the sum of the terms of the progression
gets as close to 1/(1-x) as desired, though never reaches 1/(1-x). If I
guessed right, this is a power series.

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).

************************

David C. Ullrich
.

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