Which of the following best explains why the relation dθds=r for the arc–length formula s=rθ is valid only when the angle θ is measured in radians?
(a) Because radians are appropriate only for very small angles, so the approximation s≈rθ is treated as exact, giving a constant slope of r.
(b) Because in radian measure θ is expressed in the same length units as s, causing the units to cancel in ds/dθ and leaving the constant r.
(c) Because a radian is defined as the ratio of an arc’s length to its radius, θ has no units, so dθd(θ)=1, making dθds=r.
(d) Because radians effectively set π=180, removing the conversion factor π/180 that appears with degrees and leaving dθds=r.