Interior Angles

• June 19th 2006, 06:26 AM
Quick
Interior Angles
When I was in 6th grade I came up with an equation for finding the measure of the interior angle of any vertex to a regular polygon.

$180-\frac{360}{n}$ where $n$ is the number of sides to the polygon. My equation comes out in degrees.

But you guys are saying $\pi(1-2/n)$ which comes out in radians.

Wouldn't it just be easier to say $180-\frac{360}{n}$?

P.S. I know the equations are the same, one just comes out in radians and the other in degrees; My main question is what is the point of radians?
• June 19th 2006, 07:15 AM
ThePerfectHacker
Quote:

Originally Posted by Quick
P.S. I know the equations are the same, one just comes out in radians and the other in degrees; My main question is what is the point of radians?

A scientist would say that it is because it reduces the size of numbers or something like that, (stupid reason).

The mathematical reason is more elegant. In calculus there is relationship because sine and cosine. We say that the derivative of sine is cosine. Now this is ONLY true when the angle is measured in degrees.
$\mbox{derivative of }\sin x=\cos x$
And, in degrees,
$\mbox{derivative of }\sin x^o=\frac{\pi}{180}\cos x^o$
Which one is simpler? The second one of course.
• June 19th 2006, 08:20 AM
CaptainBlack
Quote:

Originally Posted by Quick
When I was in 6th grade I came up with an equation for finding the measure of the interior angle of any vertex to a regular polygon.

$180-\frac{360}{n}$ where $n$ is the number of sides to the polygon. My equation comes out in degrees.

But you guys are saying $\pi(1-2/n)$ which comes out in radians.

Wouldn't it just be easier to say $180-\frac{360}{n}$?

P.S. I know the equations are the same, one just comes out in radians and the other in degrees; My main question is what is the point of radians?

The reasons I would recommend using radians to express angles are:

1. It is a more natural measure of angle in Mathematics, some of the
reasons for this have been given by PH, but there are others as well.

2. One of the common practical uses for trig functions is in scientific
and engineering computing, where the trig functions by default use
radian measure. If you habitually use degrees you will end up making
mistakes if you ever get to write scientific code. One common
consequence of such errors are that people die (when you bridges
fall down) or your Mars probe gets lost.

(same argument applies to using SI units rather than customary)

RonL
• June 19th 2006, 09:25 AM
Soroban
Hello, Quick!

Here's how I've tried to explain radians and natural logs to my students.

Someone (a Babylonian?) divided a circle into 360 equal parts and named them "degrees".
. . Why three-hundred-and-sixty? . . . Who cares? . Does it matter?

On another planet, say, Gamma Hydra IV in the Delta Quadrant,
. . the inhabitants might divide a circle into 500 parts and call them " $\blacktriangledown\rightsquigarrow\ddots$"
. . (nearest translation).

These angle measures are totally arbitrary.
We can create our own system: $1\text{ circle} = 137\text{ gzorns}$

In every case, when we study Calculus, we must insert an "adjustment factor" into the problem
. . to specify which angle measure we are using.

Well, almost every case . . . If we use radians, no adjustment is required.

What's so natural about using such an ugly base: 2.718281828459045 . . . ?

The number is transcendental (like $\pi$) and goes on forever, never repeating.
Who in their right mind would want to raise it to some power?

In Calculus, logarithmic problems also require an "adjustment factor"
. . to specify the base we are using . . . except when using base $e$.

With natural logs (base $e$), no adjustment is required.
So the ugly number $e$ is also built into Calculus.

Hence, they are called natural logarithms. .(Now you know why!)

It appears that $\pi$ and $e$ are woven into the very fabric of Mathematics.
These two numbers turn up in the most unexpected places.
. . For example: . $e^{i\pi} + 1\:=\:0$
• June 21st 2006, 05:20 AM
topsquark
One or two Physicists might even say that the reason that radians are preferred is because they are essentially unitless: The measure of an angle in radians is distance/distance (from the circle equation $s = r \theta$ where s is the arc length described by a central angle theta in a circle of radius r). Effectively then radians are unitless, so we can drop the unit.

(I have investigated what happens when you DON'T drop the radian unit in Physics...the results are fascinating! To me anyway. :) )

-Dan