Advantages of Radian Measurements

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Discussion Overview

The discussion revolves around the advantages of using radians over degrees in various mathematical and physical contexts. Participants explore the implications of using these two units of angular measurement, touching on their applications in trigonometry, calculus, and practical scenarios such as machinery and surveying.

Discussion Character

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

Main Points Raised

  • Some participants propose that radians are more specific to the actual circle being measured, as they relate directly to the circle's dimensions, while degrees are seen as an arbitrary division.
  • One participant notes that radians simplify calculations in applied physics, particularly in determining arc lengths using the formula \(2\pi r\).
  • Another participant mentions that radians are necessary for the correctness of trigonometric functions and calculus, highlighting that certain derivatives only hold true when angles are measured in radians.
  • There is a suggestion that the number 360 for degrees may be linked to the Babylonian base 60 number system, with some participants also noting its divisibility by several smaller numbers as a practical advantage.
  • One participant expresses a preference for gradients, suggesting they should be used more often, although this view is not widely discussed.
  • Participants confirm the accuracy of the conversion formulas between degrees and radians provided in the initial post.

Areas of Agreement / Disagreement

Participants generally agree on the advantages of radians in specific contexts, particularly in relation to trigonometric functions and applied physics. However, there is no consensus on the overall superiority of radians over degrees, as some participants highlight the practical benefits of using degrees in certain applications.

Contextual Notes

Some assumptions about the relationship between radians and the geometry of circles remain unexamined, and the discussion does not resolve the historical origins of the number 360 or the implications of using gradients.

Who May Find This Useful

This discussion may be useful for students and professionals in mathematics, physics, engineering, and related fields who are exploring the use of angular measurements in their work or studies.

_Mayday_
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Hey,

Simple questions, and hopefully I can explain my query a bit better than others I have made. I'm just trying to think of any advantages for using radians instead of degrees? I know/think that degrees are an arbitrary unit but cannot think of any reasons for using 360.

1. What are the advantages of using radians?

2. If degrees are an arbitrary unit then, where did the number 360 come from? Is it some how related to time, with 60 being a multiple of 360.

3. Am I correct in saying that to convert 1 degree to radians I would use, 1 x π/180, and if converting 1 radian to degrees, I would use 1 x 180/π

Thanks.
 
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Heuristically, if you have a definite radius of rotation, use radians; if you don't, use degrees. Thus, in rotating machinery, radian usage makes all the formulas much easier and, in surveying, degree usage makes the shots much easier.

As to why 360, many folks think the Babylonians are to blame. They used a base 60 number system. Others have noted that many ancient folks (and some modern ones) think 12 is a magical number.

I personally think gradients (that's the G in the DRG button of your calculator) got a bad rap and should be used more.

And, yes, your conversions are accurate.
 
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Would it be true in saying that using radians is more specific to the actual circle you are measuring? As it is directly related to the circle, as it uses other parts of it, almost as a ration to determin it? Where as the degrees system is only reliant on dividing the circle into 360 pieces.

Am I correct in saying that to convert 1 degree to radians I would use, 1 x π/180, and if converting 1 radian to degrees, I would use 1 x 180/π
 
Radians are nice because there are 2*pi of them in a circle, and 2*pi*r is the circumference of a circle... that means that you can multiply the number of radians in an angle by the radius to calculate the length of the arc for any slice of a circle, which turns out to be useful quite often in applied physics problems.
 
360 is great number. and what makes it great is the fact that it is divisible by 2,3,4,5,6,8...(many small and useful numbers)
 
Well I think I'll be using them quite soon both in mechanics and in physics, so it's nice to have a little head start thanks.
 
Radians are required for trigonometric functions and calculus. For example
\frac{d}{dx}\sin{x^2} = 2x\cos{x^2}
is only true if x is in radians.
 
Yes, I think that we are moving onto more advanced trigonometry within the next few weeks, and I have also heard the teacher mention them.
 

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