Your two questions are inter-related and I'll go for the second one first.
The angle one radian is defined to be the angle when the angle theta defines
an arc equal to the radius of the circle. Without loss of generality, let's
use meters as the length unit for the circumference of our circle. Then the
defining equation is:
Theta (as a radian angle) = s (meters) / r (meters).
Here the meters units cancel out and we are left with a pure number for the
measure of the angle with no units. If the arc length and radius are equal
then
Theta = 1
The degree, however is defined as 1 / 360 part of one complete revolution
about the circle. That is, one degree is the angle at which C / 360 meters
along the circumference of the circle is covered during the revolution (or
orbit) about the circle. In other words, one degree is (1 / 360) * C meters
and is tied directly to the arc length , s.
The radian by definition is the ratio of two lengths. The degree by
definition is defined as a fractional piece of the circumference , C.
When you write, for example, 45 degrees = pi / 4 radians you are saying 45
degrees corresponds to an arc length / radius ratio that is (1/8) the ratio
of the circumference to the radius of the circle.
The conversion 45 degrees * (pi / 180 degrees) represents a cancellation of
the degrees (meters) units and leaves a pure number as the pi here has no
units attached.
Using s = r * theta with theta in degrees has a unit imbalance between the
two sides as both r and theta would carry a meters unit of some kind.
David Beach
Labette Community College
Parsons, KS
> ----------
> From: Martin Kalmar[SMTP:MKalmar@fcc.cc.md.us]
> Sent: Tuesday, November 02, 1999 1:48 PM
> To: DavidB@labette.cc.ks.us
> Cc: mathedcc@archives.math.utk.edu
> Subject: RE: [MATHEDCC] Why radians?
>
>
>
> >>> David Beach <DavidB@labette.cc.ks.us> 11/01/99 01:37PM >>>
> Here is the way I was taught to think about it:
>
> If you use degrees in equations like s = r * theta for arc length on a
> circle, your answers will always be off by a factor of ( pi / 180 ).
>
> "But if you were measuring theta in degrees you wouldn't use that
> formula????"
>
> In
> calculating frequencies for motion the same phenomenon also occurs.
> Radians
> are truly unitless measures of a fractional part of the arc of the circle
> to
> the total circumference. In time units they are a fractional part of the
> time needed to complete part of a rotation along the arc on a circle
> compared to the total period of one complete rotation about the circle.
> There is truly no unit of measure "radian". It is a pure number.
> Degrees,
> however, are a unit of measure and conversions must be used to remove the
> units.
>
>
> "Please explain this last bit some more to me. I've heard people say it
> many times but I've never understood it. Radians are pure numbers if you
> choose to ignore the unit, but why can't you do the same with degrees?"
>
>
> David Beach
> Labette Community College
> Parsons, KS
> > ----------
> > From: Martin Kalmar[SMTP:MKalmar@fcc.cc.md.us]
> > Reply To: Martin Kalmar
> > Sent: Monday, November 01, 1999 11:22 AM
> > To: Mathedcc@archives.math.utk.edu
> > Subject: [MATHEDCC] Why radians?
> >
> > When students ask (not often enough) why do we need to measure angles in
> > radians, I know of no other answer than it makes things very convenient
> in
> > calculus. This is fine for my calculus students who can see what the
> > derivatives of trig functions would look like without radians, but what
> > other answer can I give my pre-calculus students?
> >
> > Does anyone know of other good reasons for using radians?
> >
> > Does anyone know the history of radian measure? Who first used it and
> why?
> >
> > Martin Kalmar
> > Frederick Community College
> > Frederick, Md.
> > mkalmar@fcc.cc.md.us
> >
> >
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