Chord (geometry)

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A chord of a curve is a geometric line segment whose endpoints both lie on the curve. A secant or a secant line is the line extension of a chord.

The red line is a chord.

[edit] Chords of a circle

Further information: Chord properties

Among properties of chords of a circle are the following:

Chords are equidistant from the center if and only if their lengths are equal.
A chord's perpendicular bisector passes the center.
If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).

[edit] Chords in trigonometry

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the Chord function for every 7.5 degrees.

The chord function is defined geometrically as in the picture to the left. The chord of an angle is the length of the chord between two points on a unit circle separated by that angle. By taking one of the angles to be zero, it can easily be related to the modern sine function:

$\mbox{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} = 2 \sqrt{\frac{1-\cos \theta}{2}} = 2 \sin \frac{\theta}{2}$

The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, not extant, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones:

Name	Sine-based	Chord-based
Pythagorean	$sin 2 θ + cos 2 θ = 1$	$\mbox{crd}^2 \theta + \mbox{crd}^2 (180^{\circ} - \theta) = 1$
Half-angle	$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}}$	$\mbox{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mbox{crd}(180^{\circ} - \theta)}$

The half-angle identity greatly expedites the creation of chord tables. Ancient chord tables typically used a large value for the radius of the circle, and reported the chords for this circle. It was then a simple matter of scaling to determine the necessary chord for any circle. According to G. J. Toomer, Hipparchus used a circle of radius 3438' (=3438/60=57.3). This value is extremely close to $180 / π$ (=57.29577951...). One advantage of this choice of radius was that he could very accurately approximate the chord of a small angle as the angle itself. In modern terms, it allowed a simple linear approximation:

$\frac{3438}{60} \mbox{crd}\ \theta = 2 \frac{3438}{60} \sin \frac{\theta}{2} \approx 2 \frac{3438}{60} \frac{\pi}{180} \frac{\theta}{2} = \left(\frac{3438}{60} \frac{\pi}{180}\right) \theta \approx \theta$