弦 定義
A chord of a circle is a straight line segment on the interior of a circle whose endpoints both lie on that circle. The infinite line extension of a chord is a secant line, or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word chord is derived from the Latin term chorda meaning bowstring. The term is also used in graph theory, where a cycle chord of a graph cycle C is an edge not in C whose endpoints lie in C.
界
圓的和弦的性質如下:
當且僅當它們的長度相等時，和弦到中心的距離是等距的。
相等的弦從圓心被相等的角度對著。
通過圓心的弦稱為直徑，是最長的弦。
如果弦 AB 和 CD 的延長線（割線）相交於點 P，則它們的長度滿足 AP·PB = CP·PD（點的冪定理）。
橢圓
橢圓的一組平行弦的中點是共線的。
三角學
和弦在三角學的早期發展中被廣泛使用。由 Hipparchus 編制的第一個已知三角表將每 7.5degrees 的弦函數值製成表格。公元二世紀，亞歷山大的托勒密在他的天文學著作中編制了一個更廣泛的和弦表，以半度為增量給出了從 1/2 度到 180 度的角度的弦值。圓的直徑為 120，弦長精確到整數部分後的兩位以 60 為基數的數字。
The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: crd θ = √ (1 – cos θ)2 + sin2 θ = √ 2 – 2cos θ = 2 sin(θ⁄2).
最後一步使用半角公式。就像現代三角學建立在正弦函數上一樣，古代三角學建立在和弦函數上。據說喜帕恰斯寫了一部十二卷的和弦作品，現在都丟失了，所以大概對它們有很多了解。在下表中（其中 c 是弦長，D 是圓的直徑），弦函數可以滿足許多類似於現代著名的恆等式：
姓名 |
基於正弦 |
基於和弦 |
畢達哥拉斯 |
sin2 θ + cos2 θ = 1 |
crd2 θ + crd2 (π - θ) = 4 |
半角 |
sin θ⁄2 = ± √ 1 - cos θ⁄2 |
crd θ⁄2 = ± √ 2 - crd(π - θ) |
Apothem (一) |
c = 2√ r2 - a2 |
c = √ D2 - 4a2 |
角度 (θ) |
c = 2r sin(θ⁄2) |
c = D⁄2crd θ |
相關定義
來源
“Chord.” From Wolfram MathWorld, mathworld.wolfram.com/Chord.html.
“Chord (Geometry).” Wikipedia, Wikimedia Foundation, 7 May 2020, en.wikipedia.org/wiki/Chord_(geometry).