家 ❯ 全部 定义 ❯ 几何学 ❯ 三角学 ❯ 单位圆 定义
单位圆 定义
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.
这个定义的另一个直接结果是能够用很少的计算显式地写出位于单位圆上的几个点的坐标。例如上图中A、B、C、D点分别对应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）。类似地，此方法可用于查找与 π⁄2 的整数倍数以及由 half 获得的许多其他角度相关联的三角函数值-angle、双角和其他多角公式。
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.
从另一个角度来看，单位圆在庞加莱双曲圆盘模型和双曲几何。在这两种模型中，双曲平面都被视为开放的单位圆盘，其中单位圆表示无限 极限点 sequences的集合在 ℍ2。
来源
“Unit Circle.” From Wolfram MathWorld, mathworld.wolfram.com/UnitCircle.html.