For theoretical applications, the radian is the most common system of angle measurement. Radians are common unit of measurement in many technical fields, including calculus. The most important irrational number π plays a vital role in radian measures of angles.
Let us introduce the radian measure of an angle.
The radian measure of an angle is the ratio of the arc length it subtends, to the radius of the circle in which it is the central angle.
Consider a circle of radius r. Let s be the arc length subtending an angle θ at the center.
θ = arc length/radius = s/r radians
s = rθ
1. All circles are similar. Thus, for a given central angle in any circle, the ratio of the intercepted arc length to the radius is always constant.
2. When s = r, we have an angle of 1 radian. Thus, one radian is the angle made at the center of a circle by an arc with length equal to the radius of the circle.
3. Since the lengths s and r have same unit, θ is unitless and thus, we do not use any notation to denote radians.
θ = 1 radian measure, if s = r
θ = 2 radian measure, if s = 2r
Thus, in general θ = k radian measure, if s = kr.
Hence, radian measure of an angle tells us how many radius lengths, we need to sweep out along the circle to subtend the angle θ.
5. Radian angle measurement can be related to the edge of the unit circle. In radian system, we measure an angle by measuring the distance travelled along the edge of the unit circle to where the terminal side of the angle intercepts the unit circle.
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