Re: A tough question about elementary mathematics

The poster formerly known as Colleyville Alan wrote:
<mariano.suarezalvarez@xxxxxxxxx> wrote in message
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The poster formerly known as Colleyville Alan wrote:
<drmwecker@xxxxxxxxx> wrote in message
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I think an important point has been missed:

In calculus and in analysis in general, we wish to have functions of a
real variable. That is, we wish the inputs to be plain old real
numbers.

In truth, we don't even need "radians" for this, and I would argue (as
I do in my teaching) that radian measure actually is a needless
mystification.

To define sin(t), where t is a real number, one winds from (1,0) on the
unit circle in the appropriate direction, arrives at a point (x, y) on
the unit circle, and takes sin(t) = the y-coordinate we just obtained.

I was trying to make a similar point, but did not think through the
matter
very well and ended up making a silly statement. But you have made the
point that I was unable to articulate, that a real number goes into the
sine
function and a y-coordinate comes out. If it so happens that the real
number is 0 then the sine is 0 if the real number is equal to pi/2 the
sine
is 1 and if the real number is equal to pi/4, the sine is srt(2)/2.

A better way of stating this is from Wikipedia. The following text gets
at
the point I was trying to make: "Although the radian is a unit of
measure,
anything measured in radians is dimensionless. This can be seen easily in
that the ratio of an arc's length to its radius is the angle of the arc,
measured in radians; yet the quotient of two distances is dimensionless."

Radians are *not* unit of measure. Angles measured in radians
*are* numbers. Pure numbers.

The point I was trying to make was that they were dimensionless. Perhaps
the Wikipedia article should have rephrased the sentence to read "although
the radian is often thought of as a unit of measure, it is dimensionless and
anything measured in radians is also dimensionless".

Of course, degrees, as you pointed out, are likewise dimensionless. Since
that is true, I have a question. I know that radians are used in calculus
rather than degrees because of the simplified properties of the derivatives.
But is there another reason to prefer radians to degrees?

When they teach the concepts of sine and cosine in trigonometry, they
usually start out with right-angle calculations and use definitions like
opposite/hypotenuse. But is the actual sine function for any x actually
just an extension of the right-angle pythagorean calculations or is there a
more formal way in which the function is defined and for which radians are a
more sensible input than degrees?

The standard power series representations of sine and cosine are very
neat when radians are used, and ugly otherwise. These series are of
course a consequence of the derivative properties (or the other way
around depending on how you look at it). The series are also closely
related to the series for exp(x), which leads to a host of important
results such as the famous exp(ix) = cos x + i(sin x).

I guess the key thing is that sine and cosine are very important "pure
mathematical" functions, irrespective of any geometrical meaning. It
just so happens that when a geometrical meaning is applied to these
functions we need to measure angles in radians.

.

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