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What is a "Radian"?


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#1 Xeon_Boy   Members   -  Reputation: 122

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Posted 09 July 2001 - 03:13 AM

Hey pals! I know what is a radius, but what exactly is a Radian? Like the explanation in English? Thanks!!!!! :-D "The feeling of mastering and understanding hard stuff in Game Programming is just like the feeling u get when u perform an Air-Walk in the basketball court, soaring.....and everyone''s watching in awe......."

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#2 masterg   Members   -  Reputation: 100

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Posted 09 July 2001 - 03:40 AM

A radian is a unit of angular measure that is equal to the angle subtended at the center of a circle by an arc which is equal in length to the radius of the circle

#3 ECKILLER   Members   -  Reputation: 122

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Posted 09 July 2001 - 03:43 AM

A radian is simply a unit of measurement for an angle. For example, 2*PI radians measures an length of a unit circle. PI measures the length of half the unit circle.

Degrees are the same deal but divided into 1/360th of a circle.

They both are simply units for measuring angles. You can choose either and convert either.

Remember how we have differant units of measure for lengths, meters, centimeters etc. We have differant units of measurment for angles to.

P.S. Its best to see a diagram in a math book to help you understand, just understand now that a radian is simply another way to measure an angle.

ECKILLER

#4 Anonymous Poster_Anonymous Poster_*   Guests   -  Reputation:

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Posted 09 July 2001 - 03:44 AM

yeah, basically, its a measure of arc length

1 radian is equal to 180/pi degrees.

hope thats simpler for you.

Nomad

#5 Shadwdrak   Members   -  Reputation: 122

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Posted 09 July 2001 - 04:05 AM

A Radian is a unit of angle measure, like a degree. There are 2PI (2x 3.14159...) Radians in one complete circle, just like there is 360 degrees in one circle. When you take your first trig class you will become very familiar with them. Radians simplify many problems, and are sometimes much easier to work with then degrees.

Here are some simple degree -> Radian equalities.
DEGREES RADIANS
360 2PI
270 3PI/2
180 PI
90 PI/2

I hope this helps some.

-Will

#6 Beer Hunter   Members   -  Reputation: 712

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Posted 09 July 2001 - 08:43 PM

The main reason for radians comes from the actual formulas for the trig functions.

sin(x) = x - x^3/3! + x^5/5! - x^7/7!...
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6!...

If you consider the unit circle, where the length along the arc of the circle is equal to the angle in radians, then you can see why that suits the trig formulas so well...

#7 grhodes_at_work   Moderators   -  Reputation: 1361

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Posted 10 July 2001 - 05:19 AM

Beer Hunter makes a good point. Radians can be considered the *natural* unit for representing angles, while degrees and revolutions or cycles are *intuitive* units for representing angles.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

#8 Shannon Barber   Moderators   -  Reputation: 1390

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Posted 11 July 2001 - 02:06 PM

I always found gradients more intuitive than degrees.... why 90 in a right angle? Doesn''t 100 make more sense?

Magmai Kai Holmlor
- The disgruntled & disillusioned


#9 Dean Harding   Members   -  Reputation: 546

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Posted 11 July 2001 - 03:10 PM

Why did they choose 360 degrees for a complete circle? Does anyone know?


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#10 Anonymous Poster_Anonymous Poster_*   Guests   -  Reputation:

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Posted 11 July 2001 - 03:26 PM


Probably because it is evenly divisible by so many factors:

2 x 2 x 2 x 3 x 3 x 5 = 360

I don''t know when 360 was first in common use, but I think it had something to do with numerology.

Bottom line is they had to pick something.

The thing is, if you don''t like 360, you can divide your circle into as many "degrees" as you want. Personally, in my programming, I usually use 3600 "degrees" in a circle.




#11 Timkin   Members   -  Reputation: 864

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Posted 11 July 2001 - 03:33 PM

On the point about one radian being the angular distance subtended by an arc of length 1 on the unit circle... as someone mentioned above... it doesn't have to be a unit circle. The funny thing about circles is that 1 radian is the angular distance subtended by an arc of length equal to the raduis, as mentioned by materg.

Oh, and Beer Hunter, those are NOT actual formulas for sine and cosine, but rather they are Power Series approximations. Exact formulas for sine and cosine can only be written using Complex exponential functions.

quote:
Original post by Dean Harding
Why did they choose 360 degrees for a complete circle? Does anyone know?



The 'Grooved Ware' people used (some 5000-10000 years ago) a system of measurement for time, distance and angle that were all based on the same base number set that had 366 'degrees' in a full circle, 60 'minutes' per 'degree' and 6 'seconds' per 'minute'. Of course, these 'degrees', 'minutes' and 'seconds' were not of the same length as the ones we use, hence the quotes to indicate this.

Using these numbers, then the distance subtended on the Earth's surface by 1 'second' of rotation of the Earth is 366 Megalithic yards (called a Megalithic Mile). From here you can see that distance can be computed in terms of rotation angle subtended on the surface of the Earth and this corresponds to a time taken to rotate through that angle/distance. This was a fantastically beautiful system of measurement.

Unfortunately, (I'm not sure but seem to recall that it was caused by a merging of the Zedokite and Cannonite religions) this number system was abolished in favour of the 360, 60, 60 system of time and angle, which was far easier to deal with. Unfortunately, this meant that distance was 'knocked out' of the combined system and hence we now have a rather ad hoc distance measurement scheme (two in fact, Imperial and Metric). If you want more information about the Pre-History of Science, check out "Urial's Machine". I cannot recall the authors right now, but will chase them up if needs be (I've loanded my copy to a friend... great book!!!!).

Cheers,

Tim

Edited by - Timkin on July 11, 2001 10:36:19 PM

#12 adtheice   Members   -  Reputation: 122

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Posted 11 July 2001 - 03:41 PM

wow I read the whole post and I got to say thanks for the free lobotomy!

Just kidding you guys realy know alot about MANY different things, and I thought I knew to much for my own good!


Edited by - adtheice on July 11, 2001 10:43:29 PM

#13 CGameProgrammer   Members   -  Reputation: 640

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Posted 11 July 2001 - 03:43 PM

You people are making radians far more complicated than they are.

A radian is equal to the circle''s radius. That''s all. It''s not an arbitrary measurement, nor was it chosen to simplify the math equations. It''s just said to be equal to a circle''s radius. Thus, since the circumference of a circle is 2*pi*Radius, there are 2*Pi radians in a circle.

~CGameProgrammer( );



#14 Xeon_Boy   Members   -  Reputation: 122

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Posted 11 July 2001 - 09:45 PM


That''s the thing I wanna hear, pal!!!!!!!! WHOA!!!!!!!!!!!!!! How simple and easy to understand, pal!!!!!!!!!!!!!!!!!!! :-D HEE-HAW!!!!!!!!!

So, that means if the radius is 4 , then the Radian is also 4 too, right? :-D THAnKS!!!!!!!!


"The feeling of mastering and understanding hard stuff in Game Programming is just like the feeling u get when u perform an Air-Walk in the basketball court, soaring.....and everyone''s watching in awe......."

#15 grhodes_at_work   Moderators   -  Reputation: 1361

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Posted 12 July 2001 - 03:56 AM

Now Xeon_Boy, lets be nice about this.

CGameProgrammer, you did write an incorrect thing when you said "A radian is equal to the circle''s radius." Not so. I think you may have written this carelessly. You actually disproved your own statement when you took the Radius out of your equation "2*pi*Radius" to arrive at 2*pi radians in a circle.

In actuality, the other explanations here are correct. Radians really do exist, as Beer Hunter said, because of the Taylor series expansions. These particular expansions are called the Maclaurin expansions:

http://www.xrefer.com/entry.jsp?xrefid=645374

The term radian seems to have appeared first in 1873 with regard to these expansions:

http://sunsite.utk.edu/math_archives/.http/hypermail/historia/mar00/0012.html

Now, as to the formula 2*pi*Radius for the circumference of a circle, well that''s just a side effect really, since the points on a circle can be constructed *using* 1 full wave of the sin(x) and cos(x) functions, which have a *wavelength* of 2*pi radians.



Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

#16 Timkin   Members   -  Reputation: 864

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Posted 12 July 2001 - 02:38 PM

quote:
Original post by grhodes_at_work

Radians really do exist, as Beer Hunter said, because of the Taylor series expansions. These particular expansions are called the Maclaurin expansions:



Radians don''t exist ''because'' of the power series approximation, but rather the power series approximation is possible because of the nature of the radian measure.

It is certainly the case the the unit angle was used long before it was called a radian and long before a Maclaurin series was used to approximate trigonmetric functions of functions of unit angles.

...on a similar vein. While it is widely believed that Hipparchus was the founder of trigonometry in the 2nd century BC, it is more than likely that he, like Pythagorus, got his information from more ancient texts (scrolls). As some circumstantial evidence, if you measure the distance around the base of the great pyramid (Pyramid of Khafra) and divide it by its height, you get a number that matches 2*pi to several decimal places. If you perform the same calculation for the Pyramid of the Sun in South America (Tiahanucan (sp?) I think) you get 4*pi. I also seem to recall that the angle subtended by an edge of the Great Pyramid with the base is equal to the unit angle (1 radian). These pyramids were purposefully built to encapsulate these numbers. It is structurally more difficult to build a square based pyramid with these properties than it is to build a uniform sqare based pyramid (45 degree sides).

Regards,

Tim

#17 grhodes_at_work   Moderators   -  Reputation: 1361

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Posted 12 July 2001 - 03:40 PM

quote:
Original post by Timkin
[quote]Original post by grhodes_at_work

Radians really do exist, as Beer Hunter said, because of the Taylor series expansions. These particular expansions are called the Maclaurin expansions:



Radians don''t exist ''because'' of the power series approximation, but rather the power series approximation is possible because of the nature of the radian measure.

It is certainly the case the the unit angle was used long before it was called a radian and long before a Maclaurin series was used to approximate trigonmetric functions of functions of unit angles.


Heh, heh...your statement is more accurate. Point taken!

quote:
Original post by Timkin
...on a similar vein. While it is widely believed that Hipparchus was the founder of trigonometry in the 2nd century BC, it is more than likely that he, like Pythagorus, got his information from more ancient texts (scrolls). As some circumstantial evidence, if you measure the distance around the base of the great pyramid (Pyramid of Khafra) and divide it by its height, you get a number that matches 2*pi to several decimal places. If you perform the same calculation for the Pyramid of the Sun in South America (Tiahanucan (sp?) I think) you get 4*pi. I also seem to recall that the angle subtended by an edge of the Great Pyramid with the base is equal to the unit angle (1 radian). These pyramids were purposefully built to encapsulate these numbers. It is structurally more difficult to build a square based pyramid with these properties than it is to build a uniform sqare based pyramid (45 degree sides).



I actually love this kind of historical study, ancient civilizations, ancient astronauts and stuff. Maybe they didn''t call them by the *term* "radians" though! Lets not get into an argument about how old the pyramids in Egypt are, though!

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

#18 grhodes_at_work   Moderators   -  Reputation: 1361

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Posted 12 July 2001 - 03:41 PM

quote:
Original post by Timkin
Radians don''t exist ''because'' of the power series approximation, but rather the power series approximation is possible because of the nature of the radian measure.

It is certainly the case the the unit angle was used long before it was called a radian and long before a Maclaurin series was used to approximate trigonmetric functions of functions of unit angles.



Heh, heh...your statement is more accurate. Point taken!

quote:
Original post by Timkin
...on a similar vein. While it is widely believed that Hipparchus was the founder of trigonometry in the 2nd century BC, it is more than likely that he, like Pythagorus, got his information from more ancient texts (scrolls). As some circumstantial evidence, if you measure the distance around the base of the great pyramid (Pyramid of Khafra) and divide it by its height, you get a number that matches 2*pi to several decimal places. If you perform the same calculation for the Pyramid of the Sun in South America (Tiahanucan (sp?) I think) you get 4*pi. I also seem to recall that the angle subtended by an edge of the Great Pyramid with the base is equal to the unit angle (1 radian). These pyramids were purposefully built to encapsulate these numbers. It is structurally more difficult to build a square based pyramid with these properties than it is to build a uniform sqare based pyramid (45 degree sides).



I actually love this kind of historical study, ancient civilizations, ancient astronauts and stuff. Maybe they didn''t call them by the *term* "radians" though! Lets not get into an argument about how old the pyramids in Egypt are, though!

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.



Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

#19 Dean Harding   Members   -  Reputation: 546

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Posted 12 July 2001 - 03:54 PM

quote:
Original post by Timkin
The ''Grooved Ware'' people used (some 5000-10000 years ago) a system of measurement for time, distance and angle that were all based on the same base number set that had 366 ''degrees'' in a full circle, 60 ''minutes'' per ''degree'' and 6 ''seconds'' per ''minute''. Of course, these ''degrees'', ''minutes'' and ''seconds'' were not of the same length as the ones we use, hence the quotes to indicate this.




Wow! That''s a really great explanation! I love little bits of trivia like this.

Hehe, Megalithic Mile - it sounds very science fiction


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#20 Timkin   Members   -  Reputation: 864

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Posted 12 July 2001 - 08:48 PM

quote:
Original post by grhodes_at_work

Lets not get into an argument about how old the pyramids in Egypt are, though!




No argument! They''re at least as old as 10450 BC give or take about 50 years... that''s the year in which there''s a natural astronomical (no, not astrological) alignment that matches the geographical alignment of many monolithic engineering feats of the era.

But more conclusively, there is much evidence to suggest that the Great Pyramid was already thousands of years old when Khafra had his face carved into the Sphinx around 2500 BC.

Additionally, the pit the Sphinx is carved into shows precipitation erosion only possible around 8000 years ago... and the direction the Sphinx is facing happens to align perfectly with the point on the horizon that the sun rises at on the first day of the age of Leo (i.e., the first time the suns precession causes it to rise in the constellation of Leo...and remember that the Sphinx is a lion)... and the year the sun first rose in Leo was... 10450 BC (give or take about 5 years)!!!

Freaky huh???!!!

Cheers,

Tim




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