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## Angle in Degrees?

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taipan_snake
Posts: 18
Joined: July 29th, 2009, 7:14 am
Location: Sydney, Australia

### Angle in Degrees?

I think it would be much more convenient if the "Angle=" thing could be done in degrees. I know, personally, it's a hassle to try to convert degrees to radians.
jsmorley
Developer
Posts: 22551
Joined: April 19th, 2009, 11:02 pm
Location: Fort Hunt, Virginia, USA

### Re: Angle in Degrees?

taipan_snake wrote:I think it would be much more convenient if the "Angle=" thing could be done in degrees. I know, personally, it's a hassle to try to convert degrees to radians.
Hi, and welcome to Ask Dr. Math.

All angle measures can be said to be dimensionless. A radian is the
ratio of an arc length to a radius, and the ratio of two lengths is
dimensionless. A degree is 180/pi radians, and the constant 180/pi is
dimensionless, so a degree is also dimensionless.

Trig functions are also by nature dimensionless. They can be defined
as ratios of sides of a right triangle. Again, the ratio of two
lengths is a dimensionless quantity (a "pure number").

Thus there is nothing surprising about the dimensional analysis of the
Taylor series for sin(x). The variable x represents a (dimensionless)
angle, so the Taylor series represents a dimensionless quantity, as
does the sine of x.

You can see that dimensional analysis isn't much use in investigating

I consider the basis for the unique significance of radian measure to
be this: The derivative of sin(x) is cos(x) (without any scale factor)
when the angle is in radians. The MacLaurin series derives from this
fact.

Without going through the proof that the derivative of sin(x) is
cos(x), we can see that the scale factor is 1 when the angle is
measured in radians. Consider a unit circle and an angle near zero.

Code: Select all

``````                         |
|
***********
*****     |     *****
***          |          ***
**             |             **
**               |               **
*                 |                 *
*                  |                  * (x,y)
*                   |         1      ---+
*                   |         -------   * s
*                    |   ------         y|*
----*--------------------+-------------------+*----
*                    |          x         *
*                   |                   *
*                   |                   *
*                  |                  *
*                 |                 *
**               |               **
**             |             **
***          |          ***
*****     |     *****
***********
|
|
``````
The sine of the small angle is y/1 = y. The radian measure of the
angle is s/1 = s. For small angles, y is approximately equal to s;
therefore the sine of the angle is approximately equal to the angle.

sin(s) ~= s

The slope (or derivative) of sin(s) with respect to theta is therefore
close to 1 for small angles. If we know that the derivative of sin(s)
is a*cos(s) for some scale factor a, then the sine of a small angle
will be close to a*cos(0) = a. Therefore the scale factor must be 1
when the angle is measured in radians.

If you used a different angle measure, such as degrees, the angle
would not be equal to s; it would be some constant (such as 180/pi)
times s. The derivative of sin(x) would then be that constant times
cos(x).

You can find discussions of the derivation of the MacLaurin series for
sin(x) in our Dr. Math archives. Here is one such discussion:

Power Series for Sine and Cosine
http://mathforum.org/dr.math/problems/mark.10.12.00.html

Does this satisfy your curiosity on this point? If you have more
questions or thoughts, we'd be glad to talk about them. It's fun to
discuss math with someone who is asking "why?"

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
taipan_snake
Posts: 18
Joined: July 29th, 2009, 7:14 am
Location: Sydney, Australia

### Re: Angle in Degrees?

O..K... then...

That was confusing...

I get the point though (I think), it wouldn't be practical to use degrees as a unit of measurement, as it is "dimensionless"...
Kaelri
Developer
Posts: 1721
Joined: July 25th, 2009, 4:47 am

### Re: Angle in Degrees?

The gist is that radians are mathematically superior because they are more pure, and can be used to identify and exploit patterns that the arbitrary 360-degree scale cannot. This makes no difference to your computer, however, so in the end there's no technical reason why this couldn't be done.

Here's an easy way to convert:

(Ï€/180) x [Angle in Degrees] = [Angle in Radians]

You can do what I do sometimes, which is plug the formula into an Excel spreadsheet, allowing you to just type in the angle in degrees and have it automatically converted.
Chewtoy
Moderator
Posts: 995
Joined: June 10th, 2009, 12:44 pm
Location: Sweden

### Re: Angle in Degrees?

Hmm...
I pondering.

Measure=Calc
Formula=X*(3.14159265/180) ;Where X is the value degrees.

Then you should just be able to get the radians, but you can write it with degrees if you want.
Unsure if you are able to link the value like that.
I don't think, therefore I'm not.
Kaelri
Developer
Posts: 1721
Joined: July 25th, 2009, 4:47 am

### Re: Angle in Degrees?

Chewtoy wrote:Hmm...
I pondering.