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Degrees are a human convention — there's nothing mathematically special about dividing a circle into 360 parts, it's a historical choice traceable to ancient Babylonian base-60 astronomy. Radians are different: a radian is defined by the geometry of the circle itself, as the angle created when an arc's length exactly equals the circle's radius, which makes a full circle equal to 2π radians (≈ 6.2832) rather than an arbitrary round number. That definition creates a direct, built-in relationship between angle and arc length (arc length = radius × angle in radians) which degrees simply don't have without an extra conversion factor.
This isn't just a stylistic preference — it has real mathematical consequences. In calculus, the derivative of sin(x) is exactly cos(x) only when x is measured in radians; if you differentiate sin(x) using degrees, an ugly conversion constant (π/180) appears and propagates through every subsequent derivative and integral involving trig functions, since the underlying reason is that the radian is the only angle unit where the small-angle approximation sin(x) ≈ x holds without a scaling correction. That's why every physics and engineering formula involving angular velocity, oscillation, or wave motion is written in radians by convention, while everyday and geometry contexts (construction, navigation, most calculators) stick with degrees because they're more intuitive to visualize without needing calculus at all.
A radian is the angle created at a circle's center when the arc length along the circle's edge exactly equals the circle's radius. Because a circle's full circumference is 2π times its radius, a complete circle equals exactly 2π radians (about 6.2832) — a value derived directly from the circle's geometry, not an arbitrary round number like 360.
It traces back to ancient Babylonian astronomy, which used a base-60 (sexagesimal) number system — 360 being 6×60, and also roughly matching the number of days in a year as ancient astronomers understood it. It's a historical and cultural convention, not a value derived from any mathematical property of circles.
Because the clean result that the derivative of sin(x) equals cos(x) only holds true when x is measured in radians. Using degrees introduces an extra conversion constant (π/180) that has to appear in every derivative and integral involving trig functions from that point forward — mathematically valid, but needlessly messy, which is why radians became the standard for calculus and physics.
Degrees remain more intuitive for everyday and geometric contexts — visualizing a 90° angle or a 45° turn doesn't require any calculus background, while radians (multiples of π) are less immediately visual for most people. Radians matter specifically in calculus, physics and engineering formulas involving angular motion, where the mathematical relationships genuinely simplify; outside those contexts, degrees remain the more practical everyday choice.