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Since 360 degrees = 2*pi radians, converting between the two systems uses the factor pi/180 (degrees to radians) or 180/pi (radians to degrees). Fluency in conversion is necessary because many real-world problems use degrees while all calculus uses radians. You should eventually recognize common conversions (30 = pi/6, 45 = pi/4, etc.) without computing.
Derive the conversion factor from the fundamental relationship. Practice converting standard angles in both directions. Build a reference table of common angles and their radian equivalents. Eventually move to instant recognition rather than calculation.
The key to converting between degrees and radians is a single foundational equivalence: one full revolution equals both 360 degrees and 2π radians. This is not a definition to memorize blindly — it follows from what radian measure means. One radian is the angle that subtends an arc equal in length to the radius. Going all the way around a circle covers an arc of length 2πr (the circumference), so a full revolution sweeps 2π radians. That's it: 360° = 2π rad.
From this single fact, all conversions follow. To go from degrees to radians, multiply by (2π / 360) = π/180. To go from radians to degrees, multiply by (360 / 2π) = 180/π. You can always rederive these factors if you forget them — just start from 360° = 2π and cross-multiply. For example, to convert 90°: multiply by π/180 to get 90π/180 = π/2 radians. To convert 3π/4 radians: multiply by 180/π to get 3 × 180/4 = 135°.
The common angles are worth internalizing as direct associations rather than computed results. 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. Once you see these often enough, you should recognize them the way you recognize that 1/4 = 0.25 — not by dividing each time, but instantly. The pattern helps: the denominator tells you what fraction of 180° (= π) you have, so π/6 is one-sixth of a half-turn (= 30°).
Notice that radian answers should generally be left as exact multiples of π, not decimal approximations. Writing π/4 is exact and clean; writing 0.785 is an approximation that loses information and makes further calculations messier. The decimal form of pi (3.14159...) is a number, not an angle unit — saying "π radians ≈ 3.14 degrees" is a category error. Fluency in this conversion is essential because calculus uses radians exclusively: the derivative of sin(x) is cos(x) only when x is in radians. Every trigonometric formula from here on assumes radian input.