Actually, maths is not my strong subject, but I'll try my best. The proof of that limit using the geometric method, like the sector area comparison, only works if the angle at the centre is exactly the arc length. That needs radians, where arc length is simply rθ. If you used degrees, you'd have messy conversion factors all over the limit and it wouldn't simplify to 1. I remember this because Kaku was telling me last week, while watering his brinjal, "Beta, in our well, the pulley's rope length for one full circle is 2πr, simple like radians! If we measured it in degrees, the calculation for how much water comes up would be so 'avakado' (complicated)!" He finds maths in farming.
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