Radians - why do we use these instead of decimal?

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Discussion Overview

The discussion centers around the use of radians versus degrees in measuring angles, exploring the reasons for the preference of radians in mathematics and calculus. Participants share their understanding of how radians are calculated, their historical context, and their implications in mathematical functions and calculus.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about how radians are calculated and seeks a formula, noting their understanding of specific angle conversions.
  • Another participant provides the formula for converting degrees to radians as radians = degrees * π/180.
  • Some participants argue that radians are based on mathematical principles, making them more natural than degrees, which they view as arbitrary.
  • It is mentioned that the derivative of sine in radians is cos(x), while in degrees it involves a constant factor, complicating calculations.
  • Some participants highlight that radians arise naturally in mathematical contexts, making them preferable for calculus and other advanced topics.
  • There is a discussion about the historical use of degrees, tracing it back to the Babylonians and their calendar, while radians have been adopted more recently in mathematics.
  • One participant notes that radians can be expressed without π, suggesting that they can be understood in decimal form as well.
  • A participant emphasizes the importance of lectures and direct communication with instructors for deeper understanding.

Areas of Agreement / Disagreement

Participants express a mix of agreement and disagreement regarding the use of radians versus degrees. While some support the use of radians for their mathematical properties, others defend the historical significance and practicality of degrees. The discussion remains unresolved with multiple competing views on the topic.

Contextual Notes

Some participants mention limitations in understanding due to reliance on calculators and the potential for confusion between different angle measurement systems, such as degrees, radians, and gradians.

Who May Find This Useful

This discussion may be useful for students learning about trigonometry and calculus, educators looking for insights into common student misconceptions, and anyone interested in the historical context of mathematical concepts.

bluebeaker
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I'm basically homeschooled in math so please forgive me if this question seems simple, but I have never been able to find a book that shows how radians are calculated. Through inspection I've sort of determined 45 degrees = pie/4, 60 degrees = pie/3, 90 degrees = pie/2, 180 = pie and so forth but I input pie into my calculator and change to degrees and it returns 200?

Is there a formula I can use to calculate radians? There is no way I can commit the tables to memory... :frown:

I can understand why before calculators, radians were easier to use, is there and reason why we continue to use radians over decimal today? Maybe if I understood them their continued use would make more sense to me.

Many thanks!
 
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degree*PI/180,higher calculus they use radians...and trig functions in programming use it.
 
The formulas you worked out are correct, and can be generalized as mentioned to be radians = degrees * pi/180.

There is a very good reason to use radians instead of any other system you can think of. For one thing, all other systems are arbitrary, radians is based on mathematical fact. The length of a curve traced along a circle of radius R for an angle theta = R*theta, if and only if theta is measured in radians.

Mathematical functions like sine and cosine for angles have series expansions. sin x = 0 where x = n*pi, not n*180. Those expansions are Taylor expansions, mathematically solid, not arbitrary.

Personally, I hate any system other than radians. Radians are natural units, degrees are man-made.
 
Really you shouldn't even think of a "radian" as a unit. When you say that an angle is \theta (in radians), you just mean that if you rotate a unit vector by that much, the length of the arc its tip traces out is \theta. So it's really dimensionless. We call them "angles" because we want to identify them as something special having to do with rotations. You'll notice that this is really the only place people use "synthetic" units in mathematics, and it's annoying!

Degrees are an arbitrary unit. I am not sure why people insist on continuing to use them. Radian measure arises naturally.

As BoTemp mentioned,

\theta = \left(\frac{180\theta}{\pi}\right)^\circ

(where \theta is the natural angle measure in radians).

If you calculator told you that \pi = 200^\circ, then your calculator is broken :-p. It is more likely, however, that you actually accidentally converted to "gradians," for which there are 400 in a circle. This is just another arbitrary unit. The conversion there is

\theta = \frac{200 \theta}{\pi} \mbox{grad},

or from degrees to gradians,

x^\circ = \frac{200 x}{180}\mbox{grad}.
bluebeaker said:
I can understand why before calculators, radians were easier to use, is there and reason why we continue to use radians over decimal today?

Actually it's quite the opposite, the degree measure was first used by the Babylonians thousands of years ago. From Wikipedia:

Wikipedia_degrees said:
The number 360 was probably adopted because it approximates the number of days in a year. Primitive calendars, such as the Persian Calendar used 360 days for a year. This was most likely due to watching stars revolve around the North Star forming a circle one degree per day. Its application to measuring angles in geometry can possibly be traced to Thales who popularized geometry among the Greeks and lived in Anatolia (modern western Turkey) among people who had dealings with Egypt and Babylon.

On the other hand, radians (or, at least, the natural angle measure, if not by that name) have only been used for a few hundred years. All mathematicians use them now, because they arise so naturally and make a lot of things more convenient.
 
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Oh, and by the way, \pi is "pi." :rolleyes:
 
360 was probably also used because of how divisible it is: into halves, thirds, fourths, fifths, sixths, etc.
 
Another point is this: if x is in radians, then the derivative of sin(x) is cos(x). If in degrees (I don't understand why you refer to degrees as "decimal") then the derivative of sin(x) is \frac{180}{\pi}cos(x).
 
Maybe he doesn't realize that radians don't have to be written out as multiples of \pi (so for example 1 = \left(\frac{180}{\pi}\right)^\circ, or 60^\circ \approx 1.0472.).
 
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we use radians in calculus to simplify the answers, i.e. in radians, sin' = cos, but not in degrees. (there would be a constant factor in there, maybe of 2pi/360?.)this is the same reason we use e^x instead of 10^x as basic exponential function.

i.e. d/dx e^x = e^x but d/dx 10^x = 10^x ln(10), where ln is inverse to the function e^x.

this is also the same reason physicists use consistent units, i.e. one unit of mass is the mass of one unit of volume of pure water at sea level at a standard

temperature? etc...

one unit of volume is a cube with sides one unit of length,...

a radian is the angle subtended by an arc whose length is one standard unit of length where the standard unit of length on a circle is of course the radius.
 
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  • #10
all those who think lecturers are useless and books hold all knowledge pay attention. of course this answer is in some books, but which ones?

i do know some cocky students who used to go to class only once, to get the lecturers reading list.
 
  • #11
Many thanks, that helps a great deal (and for the spelling correction too). I've been up 4 days (2 days straight) learning complex numbers and polar coordinates - needless to say I'm radian-mad. I've been approaching this chapter by trying to equate the radians to decimal, I guess because they make me uneasy. I understand now I'm looking at them the wrong way.

Agreed, lectures are invaluable. I'm completing this course through correspondance so I can be accepted into uni and WOW will I ever appreciate having a math prof who I can speak with. Until then this forum is a wealth of knowledge and much appreciated...
 
  • #12
i am a math prof at uni. and we are speaking. best wishes.
 
  • #13
by the way, simple appreciation for what you are taught, such as you have shown here, will get you plenty of willing teachers.
 
  • #14
Office_Shredder said:
360 was probably also used because of how divisible it is: into halves, thirds, fourths, fifths, sixths, etc.

Actually, the degree measure of angles dates back to the ancient Babylonians, who had a roughly 360-day calendar.
 
  • #15

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