The Greek chord function (usually written as crd) gives the length of the chord of a circle subtending a known central angle. In a very real sense, the chord function is all about the properties of a right-angled triangle inscribed in a semi-circle. If you had to perform the necessary calculations by hand, it would take you a very long time. What you may not know, however, is that the microprocessor inside your calculator must run a highly complex algorithm in order to compute an accurate value. Your calculator can find the sine of an angle instantly, and with a high degree of accuracy. The trigonometric sine function is the modern equivalent of the Greek chord function (which we will talk about shortly). What they didn't have was a simple formula for calculating the size of the central angle subtended by a chord of known length (or vice versa). They were certainly capable of constructing circles of a known diameter, and chords of a known length (you have probably done this at school with a pencil, a ruler and a pair of compasses). The Greeks knew that the size of the central angle subtended by a chord of a circle is proportional to the ratio of the length of the chord and the diameter of the circle. Chords of the same length subtend the same central angle.A chord's perpendicular bisector passes through the centre of the circle.Before we continue, there are a couple of things to keep in mind about chords of a circle: Until the third century BCE, however, the theorems developed by the Greeks were presented in geometric rather than algebraic terms. This of course included knowledge of the properties of chords, and angles inscribed in a circle. It is evident from what we know of the writings of classical Greek mathematicians, such as Archimedes and Euclid, that the Greeks already had an extensive knowledge of geometry. One of the greatest problems they faced was how to accurately measure angles. The branch of mathematics known as trigonometry began to emerge some three to four centuries BCE, as Greek astronomers and mathematicians searched for a consistent way of measuring and recording the movement and relative positions of various celestial bodies. Sines are described on the next page.Chord AB subtends arc AB and central angle α Sines were first used in India a few centuries after chords were first used in ancient Greece. Of course using a unit circle doesn’t avoid fractions, but we have decimal fractions which are easy to work with.Īlthough trigonometry was, and still could be, based on chords as the primary trigonometric function, a slight modification of chords, called “sines,” turns out to be more convenient. In contrast, our present-day trigonometric functions are based on a unit circle, that is, a circle of radius 1. The advantage of a large radius is that fractions can be avoided. Ptolemy used a different large fixed radius than Hipparchus. It also included aids for interpolating chords for minutes of angle. His table had chords for angles increasing from 1/2 degree to 180 degrees by steps of 1/2 degree. Later, Ptolemy (100–178 C.E.) constructed a more complete table of chords. Incidentally, his table was not in terms of degrees, but “steps”, each step being 1/24 of a circle. It was a table of chords for angles in a circle of large fixed radius. Hipparchus (190–120 B.C.E.) produced the first trigonometric table for use in astronomy.
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