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單位圓 定義
The unit circle is a circle with a radius of 1 which is centered at the origin on the x-y plane. The unit circle plays a significant role in several different areas of mathematics. In particular the functions of trigonometry are most simply defined using the unit circle. As shown in the figure below, a point p on the terminal side of an angle θ in angle standard position measured along an arc of the unit circle has as its coordinates (cos θ, sin θ) so that cos θ is the horizontal coordinate of p and sin θ is its vertical component. As a result of this definition, the trigonometric functions are periodic with period 2π.
這個定義的另一個直接結果是能夠用很少的計算顯式地寫出位於單位圓上的幾個點的坐標。例如上圖中A、B、C、D點分別對應π⁄3、3π的角度。 ⁄4、7π⁄6 和 11π&frasl ;6 弧度，由此得出 A = (1⁄2, 3⁄< sub>2), B = (-1⁄2, 1⁄2 )、C = (-3⁄2、-1⁄2) 和 D = ( 3⁄2、-1⁄2）。類似地，此方法可用於查找與 π⁄2 的整數倍數以及由 half 獲得的許多其他角度相關聯的三角函數值-angle、雙角和其他多角公式。
The unit circle can also be considered to be the contour in the complex plane defined by |z| = 1, where |z| denotes the complex modulus. This role of the unit circle also has a number of significant results, not the least of which occurs in applied complex analysis as the subset of the complex plane where the Z-transform reduces to the discrete Fourier transform.
從另一個角度來看，單位圓在龐加萊雙曲圓盤模型和雙曲幾何。在這兩種模型中，雙曲平面都被視為開放的單位圓盤，其中單位圓表示無限 極限點 sequences的集合在 ℍ2。
來源
“Unit Circle.” From Wolfram MathWorld, mathworld.wolfram.com/UnitCircle.html.