Weblog on Physics: Chords in trigonometry

Friday, 12 July 2013

Chords in trigonometry

Source: Wikipedia.

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. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part. 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. 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 $\theta$ , sin $\theta$ ), and then using the Pythagorean theorem to calculate the chord length:

$\mathrm{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} =2 \sin \left(\frac{\theta }{2}\right).$

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.

The chord function satisfies many identities analogous to well-known modern ones:

Name	Sine-based	Chord-based
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1 \,$	$\mathrm{crd}^2 \theta + \mathrm{crd}^2 (180^\circ - \theta) = 4 \,$
Half-angle	$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}} \,$	$\mathrm{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mathrm{crd}(180^\circ - \theta)} \,$
Apothem (a)	$c=2 \sqrt{r^2- a^2}$	$c=\sqrt{D ^2-4 a^2}$
Angle (θ)	$c=2 r \sin \left(\frac{\theta }{2}\right)$	$c=D \sin \left(\frac{\theta }{2}\right)$