Page:The New International Encyclopædia 1st ed. v. 19.djvu/528

* TRIGONIA. 460 TRIGONOMETRY. functions are tangent of B, symbolically written a h tan := -r; culanqent of 8, or cot B =z -- ; secant b " a ' of $, or sec B = —; and cosecant of B or cosecfl c ° = — . There are also used the functions ver- a J sine of 8, or vers B = 1 — — ; and coverstne of c e, or covers S =1 — — . The propriety of calling these ratios 'functions' of the angle 8 consists in this, that the value of any ratio depends upon the value of B. That is, in any right-angled triangle AB'C, having an acute angle B, the a' h' a' corresponding ratios — . ~< Ti (Fig. 2) are aha, equal to the ratios — . —, -r- ; and in any right- angled triangle in which the acute angle B' is not equal to 8 tlip corresponding ratios are not equal to those for B. The trigonometric func- tions as defined by the above ratios are evi- C <rf is, three -f 7wvfo, jroma, angle) . This genua, represented at the present time by only five species living in the Australian seas, was one of the most important types of lamellibranchs in the Mesozoic seas of Europe. Trigonia appeared first in the Liassic, became very abundant in the Middle and Upper Jura and the Jliddle Cre- taceous, and then in the Tertiary declined in number of species to the present time. The Mesozoic species, some of which are four or five times the size of the living species, are arranged into groups according to the markings of their shells, which are often elaborately sculptured. Consult: Louis Agassiz, Etudes critiques sur les mollusqucs fossilcs. Memoire sur les trlflonics (Neuchfitel, 1840); Lycett, "A Monograph of the British Fossil Trigonias," Palwontographical Societt/ Monographs (London, 1872-7!)) ; Von Zittel and Barrois, Traitc de paUontoIogic, part i., vol. ii. (Paris, Munich, and Leipzig, 1887). TRIG'ONOCAR'PUS (Neo-Lat., from Gk. rpiyavoi, trigOnos, triangular + Kap7r6s, kar- pos, fruit). Fossil seed pods and fruits, prob- ably of various kinds of plants, which in cross section have a triangular form. See C.iEPOi-lTH; CORD.ilTES ; CO.NIl'ER.E ; PALEOBOTANY. TRIGONOMETRIC SERIES. See Sehies. TRIGONOMETRY (from Gk. Tplyoivov, trigOnoH, triangle, from rpeis, treis, three + yuivla, gOnia, angle -f -M^rpfa, metria, measure- ment, from fiirpov, mctron, measure, from lufpety, metrein, to measure). Originally the study of triangles, especially the theory of the measurement of their sides, angles, and areas; now the measure of triangles is merely a part of the general subject. That portion of the subject which deals with the measurement of figures in a plane is called plane trigonometry, and that which deals with figures on the surface of a sphere is called spherical trigonometry. That branch of the subject which deals with the circular functions of angles is called goniometry. The pure theory of trigonometric functions, apart from their application to problems of measurement, is called analytic trigonometry. Elementary trigonometry has many useful ap- plications, as in the measurement of areas, heights, and distances. It is indispensable to the study of astronomy, physics, and the various branches of engineering. FlQ. 3. In Fig. 3 the radius OA( = OB) may be re- garded as the unit of length, hence the ratio H15- = BJVIi and sin AOBj = BiM. Similarly cosAOBi = OM, tanAOBj = AT, cotAOB, = PQ, secAOB, = OT, cosecAOBj = OP, versAOB, — MA, and coversAOBi = QR. If the angle is obtuse as AOB,, or reflex as AOB3, AOB., the The common functions of trigonometry may be functions are represented by the corresponding defined as ratios of certain sides of a right tri- lines. E.g. sin AOB3 = B3M1, tan AOBj = AT'. a The following convention of signs (see Fig. 4), angle. Thus, in the figure, the ratio - is called },o„.pver. serves to associate these values with the the sine of the angle 6, commonlv written sin 9 proper angle: Lines measured to the right of ^ " J the vertical diameter, as OM, are called positive, = — . The ratio — is called the cosine of and those to the left, as OMj, negative; lines " " * m, I, measured upward, as B M, from the horizontal the angle e, written cos«= -. The other diameter arc called positive, and those downward. dently limited to angles less than 90°, since a triangle contains but one right angle. However, the definition may be extended to angles of any size and the functions expressed by line seg- ments.