Math & Statistics · 单位圆（Unit Circle） · Updated 2026-10-25
What Is Unit Circle?
Definition
The unit circle (radius 1, centered at origin) geometrizes trigonometry: the terminal side of angle θ meets the circle at (cosθ, sinθ), with tanθ=sinθ/cosθ. Radians follow: 1 radian subtends an arc equal to the radius, and 180° = π rad. Key facts: sin/cos at 30°/45°/60° (√2/2, √3/2 family) and quadrant signs (ASTC: all positive in I, sine in II, tangent in III, cosine in IV). Circular motion, AC phase and Fourier decomposition all inherit this picture.
Formula
x = cosθ, y = sinθ; x² + y² = 1; 180° = π rad
Worked Example
At θ=60° the point is (1/2, √3/2) — cos60°=0.5, sin60°≈0.866; a clock hand sweeping 90° covers π/2 radians and arc length r×π/2.
Common Misconception
"sin²θ+cos²θ=1 is a coincidence" — it is the Pythagorean theorem projected onto the circle. And degrees never mix with radians: sin(90) in radian mode is sin(90 rad)≈0.894, not 1 — the DEG/RAD toggle is the classic exam trap.
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