Rhyme and Reason Behind Radians?

In summary, the conversation discusses the concept of radians and why they are defined as two pi per circle. The speaker suggests using tau (representing 2*pi) instead for a more intuitive understanding. The conversation also touches on the historical context of how pi was originally defined and its use in modern mathematics. However, it is unlikely that tau will be adopted in academia due to established conventions.
  • #1
1d20
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Preface: This question is a result of personal interest, and has nothing to do with any assignment.

Okay here's the deal; I've tutored trig in the past, and I've noticed one thing that a lot of folks have trouble with is the fact that there are two pi radians per circle. (It doesn't help that pi can't help but remind most of us of a deliciously full pie.)

Anyway, that got me thinking: why are radians defined such that two pi fits in each pie? We could easily redefine radians such that pi = pie, which is much more intuitive. (Don't believe me? Radian angle = S/2R. It works.)

So I'm hoping someone here is a math history buff, and can give me an answer. It'll probably be "Well this is how the first guy did it, so that's how we do it too," but I can hope for a more interesting answer.

Thanks in advance. :)
 
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  • #2
1d20 said:
Preface: This question is a result of personal interest, and has nothing to do with any assignment.

Okay here's the deal; I've tutored trig in the past, and I've noticed one thing that a lot of folks have trouble with is the fact that there are two pi radians per circle.

Google "pi is wrong" and you will see many links about this subject. There are a few threads here on Physics Forum as well.

Some people use the Greek letter tau to stand for 2*pi, and then there are exactly tau radians in a circle; half a circle is tau/2, etc. Much more sensible. Maybe this will catch on.

My theory (which I have no confirmation for) is that historically, people invented geometry and trigonometry to measure the size of their fields (agricultural fields, not algebraic ones of course!) so that the diameter of a circle is the most important measurement. Circumference/diameter was what they studied, and that's how they got pi.

However in modern times we understand circles mathematically in terms of the plane and the unit circle; and in that context, the radius is more important. And the circumference divided by the radius is unfortunately 2*pi, from which much student confusion follows.
 
  • #3
SteveL27 said:
Maybe this will catch on.

It may catch on in education circles, but unlikely in academia. We use tau as a variable in too many fields. The reason why pi never seems to be used as a variable is because we use it as a constant and it's ingrained.

Similarly, you see in high school texts:
[itex]\overline{A}[/itex] to mean the complement of a set, not its closure
the use of [itex]\mathbb{J}[/itex] to be the integers not [itex]\mathbb{Z}[/itex]
the use of cis
and so on. And students have to unlearn these conventions because no academic paper uses them.
 

FAQ: Rhyme and Reason Behind Radians?

1. What are radians and how are they different from degrees?

Radians are a unit of measurement used in mathematics and science to measure angles. They are different from degrees in that they measure the size of an angle based on the radius of a circle, rather than the number of degrees in a full circle.

2. Why are radians important in trigonometry and calculus?

Radians are important in trigonometry and calculus because they provide a more precise and universal way to measure angles. They also make it easier to perform calculations and solve equations involving angles and circular motion.

3. What is the relationship between radians and the unit circle?

Radians and the unit circle are closely related because the measurement of an angle in radians is equal to the length of the corresponding arc on the unit circle. This relationship is used to convert between radians and degrees.

4. How do radians relate to the concept of pi?

Radians are based on the number pi (π), which is the ratio of a circle's circumference to its diameter. One full rotation around a circle is equal to 2π radians. This relationship is used to convert between radians and degrees.

5. Can radians be negative or greater than 2π?

Yes, radians can be negative or greater than 2π. Negative radians indicate a clockwise rotation, while positive radians indicate a counterclockwise rotation. Radians greater than 2π represent multiple rotations around a circle.

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