When we measure angles in radians, the arc length equals the radius times the angle, so for a small x, sin x approximates the arc itself. That’s why the ratio sin x / x approaches 1. If we used degrees, the arc length relation breaks, and the limit becomes π/180. I remember this clearly because I derived it on the terrace last night, using the geometry of the neem tree’s shadow, while Appa was listening to his radio.
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