Measurements of angles in circular method

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Discussion Overview

The discussion revolves around the definition of the radian as a measure of angles, specifically the relationship between the angle in radians, the arc length, and the radius of a circle. Participants explore the reasoning behind this definition and its implications in mathematics and physics.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Exploratory

Main Points Raised

  • Some participants assert that the relationship θ in radians equals arc/radius is a definition rather than a provable statement.
  • One participant suggests that the definition of a radian is motivated by the simplicity it brings to mathematical formulas, particularly in relation to angular and tangential velocity.
  • Another participant notes that if the radian were defined differently, many mathematical formulas would require adjustments due to the constant factor involved.
  • There is a mention of the relationship between the definition of radians and the derivatives of trigonometric functions, indicating a connection to calculus.

Areas of Agreement / Disagreement

Participants generally agree that the radian is a definition and not something that can be proved. However, there are varying interpretations of the implications and motivations behind this definition, leading to a lack of consensus on the reasoning behind it.

Contextual Notes

Some assumptions about the definitions and relationships discussed may not be explicitly stated, and the implications of the radian definition on various mathematical concepts remain open to interpretation.

parshyaa
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  • Why θ in radian equals arc/radius?
I know that it can't be proved but there must be a explanation for this formula. How founder may have got this idea.
 
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parshyaa said:
  • Why θ in radian equals arc/radius?
I know that it can't be proved but there must be a explanation for this formula. How founder may have got this idea.
It's a definition. Definitions aren't proved.
One radian is the angle subtended by a sector of a circle for which the arc length of the sector is equal to the radius of the circle.
 
Mark44 said:
It's a definition. Definitions aren't proved.
One radian is the angle subtended by a sector of a circle for which the arc length of the sector is equal to the radius of the circle.
OK, 2πr/1 = 360°, can we say that here arc = 2πr and radius =1 , therefore we get 2π = 360° , this may be the reason which made founder to make it as a definition , this is just my thinking
 
parshyaa said:
OK, 2πr/1 = 360°, can we say that here arc = 2πr and radius =1 , therefore we get 2π = 360° , this may be the reason which made founder to make it as a definition , this is just my thinking

A major motivation for defining "radian" (in the standard way) is that it makes many mathematical formulas simple. If "radian" were defined differently, then many formulas that effectively have the constant factor of "1" in them would have to be rewritten with a different constant factor.

Are you familiar with the definitions of "angular velocity" and "tangential velocity" as applied to an object moving in a circle ? The definition of "radian" creates a simple relation between them.

If you have studied calculus, you can understand that the definition of "radian" creates a simple relation between trigonometric functions (like sin(x)) and their derivatives.
 

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