Equations · equation
Arc Length and Radians
Vectors & Geometry · 1714
- Equation
- s = rθ (θ in radians; 2π rad = 360°)
- Field
- Vectors & Geometry
- Symbols
- s length along the arc · r radius · θ the angle subtended, measured as arc length over radius — a pure number
- Year
- 1714
Significance
An angle is a ratio, not a count of degrees: the arc it cuts from a circle divided by the radius. That is what makes sin θ ≈ θ true for small angles, ω = v/r hold without a conversion factor, and every equation with a frequency in it come out in the right units. Roger Cotes saw in 1714 that this was the natural measure; the word 'radian' arrived in 1873.
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