TRIGONOMETRY

mathematical subject dealing with the numerical relationships among the sides and the angles of a triangle. In plane trigonometry the triangle to be studied is formed by selecting three points in a two-dimensional plane (so that all three points do not lie on the same straight line) and connecting each pair of points with a straight line segment. In spherical trigonometry the triangle to be studied is formed by selecting three points on the two-dimensional surface of a three-dimensional sphere (so that all three points do not lie on the same great circle) and connecting each pair of points with an arc of the great circle that they determine on the sphere. Spherical trigonometry arose in the early Greek period in response to a practical need for accurate information in astronomy and for efficient and systematic methods of obtaining this information. Plane trigonometry was implicitly contained in the study and methods of spherical trigonometry, but it was not until the demands of surveying, trade, and navigation of 15th-century Europe that it came to be studied and applied in its own right, quite independently of its connection with spherical trigonometry. Although spherical trigonometry was developed before plane trigonometry, it is easier to understand and appreciate spherical trigonometry if one is aware of the essentials of plane trigonometry. Plane trigonometry is ordinarily approached through the use of right triangles, although the subject actually deals with any type of triangle. In this approach, given an angle A that is between 0 and 90, a right triangle is constructed having A as one of its angles. (Since the sum of the three angles of a plane triangle is always 180, the third angle must be 90 - A.) The side opposite the right angle is labeled the hypotenuse, the side opposite angle A the opposite, and the side next to angle A the adjacent. There are six possible ratios of side lengths that can be formed, and these are used to define the six trigonometric functions of angle A: Values of the trigonometric functions of different angles have been tabulated, and the tables allow the theory of plane trigonometry to be applied to problems in such pursuits as surveying and engineering. To illustrate this, consider a building of unknown height that must be measured. If the building is perpendicular to the ground on which it stands, then a right triangle can be formed having as its vertices the base of the building, a point a measured distance away along the ground, and the top of the building. The relationship now holds in this triangle, where A is the angle of elevation from the measuring point to the top of the building. Since the distance from base is known, the angle A can be measured using ordinary surveying instruments, and tangent A can be looked up in a table of trigonometric functions, the height of the building can be found from this trigonometric relationship. Given any (not necessarily right) triangle with an angle A, one of the areas of study in plane trigonometry is to establish relationships among the trigonometric functions of angle A, and these relationships are called trigonometric identities. The most basic trigonometric identity is sine2 A + cosine2 A = 1. That is, the square of the sine of an angle plus the square of the cosine of the same angle is always equal to 1. Another area of study is the calculation of the trigonometric functions of certain angles from those of other angles. An example of this is the double-angle formula for the sine: sine 2A = 2(sine A)(cosine A). Using this formula, the sine of the angle 2A can be obtained directly from the known values for the sine and cosine of angle A. Similar double-angle formulas have been developed for the other trigonometric functions as well. The beginnings of spherical trigonometry can be traced to the work of the astronomer Hipparchus of Nicaea (180125 BC), who has come to be known as the father of trigonometry. One of the uses of astronomy in Greek and pre-Greek times was to tell the time of day or period of the year by the positions of various stars. Astronomers compiled charts of the angular measure between various stars. Hipparchus systematically studied the relationship between an arc of a great circle on the celestial sphere and its corresponding chord (i.e., the distance between two points as measured along the surface of the sphere, and the distance as measured along a straight line, respectively). The Alexandrian astronomer Ptolemy, who worked in the 2nd century AD, continued Hipparchus' work in the computation of arcs and chords of great circles. Greek trigonometry reached its height with Ptolemy, whose major work in the subject has come to be called the Almagest (The Greatest). In it, Ptolemy obtained relationships among the angles and sides of spherical triangles that are the equivalent of many basic relationships and identities of trigonometry today. the branch of mathematics concerned with specific functions of angles and their application to calculations in geometry. For example, if a right-angled triangle contains an angle, symbolized here by the Greek letter alpha, a, the ratio of the side of the triangle opposite to a to the side opposite the right angle (the hypotenuse) is called the sine of a. The ratio of the side adjacent to a to the hypotenuse is the cosine of a. These functions are properties of the angle a, and calculated values have been tabulated for many angles. They are used in obtaining unknown angles and distances from known or measured angles in geometric figures. The subject of trigonometry developed from a need to compute angles and distances in such fields as astronomy, map making, surveying, and artillery range finding. Problems involving angles and distances in one plane are covered in plane trigonometry. Applications to similar problems in more than one plane of three-dimensional space are considered in spherical trigonometry. Additional reading Discussions of trigonometry can be found in Ernest William Hobson, A Treatise on Plane and Advanced Trigonometry, 7th ed. (1928, reprinted 1957); Mary Claudia Zeller, The Development of Trigonometry From Regiomontanus to Pitiscus (1946); Herbert E. Salzer and Norman Levine, Table of Sines and Cosines to Ten Decimal Places at Thousandths of a Degree (1962); Myra McFadden, Modern Trigonometry: A Program for Self-Instruction (1965); Rudolf A. Zimmer, Basic Trigonometry with Applications in Technology (1980); Paul A. Foerster, Trigonometry: Functions and Applications, 2nd ed. (1984); Marshall D. Hestenes and Richard O. Hill, Jr., Trigonometry, 2nd ed. (1986); and Thomas W. Hungerford and Richard Mercer, Trigonometry (1992). Texts that include a treatment of trigonometry are David Eugene Smith, History of Mathematics, 2 vol. (192325, reprinted 1958); J.L. Berggren, Episodes in the Mathematics of Medieval Islam (1986), chapters 56; and W.M. Smart, Text-Book on Spherical Astronomy, 6th ed. rev. by R.M. Green (1977), chapter 1. Raymond Walter Barnard The Editors of the Encyclopdia Britannica