Again, from fig. 1, we obtain, from the properties of right-angled and of similar tri angles, the following relations between the functions: = = = tau. : R :: sin. : cos., sec. : R : cos., cot. : R :: cos. : sin., cosec. : R R : sin., and cot. : R R : tan. From these eight relations we can easily obtain any one function in terms of any other, both as regards its magnitude and sign.
The reason why the circle its radius are employed in the definition of the func tions is that we may obtain some invarial le standard by which to estimate them, for while, as the angle increases from 0° to its functions are in a state of constant change, their standard of the remains the same. For greater cation the radius is taken as and the relations become = 1, tan.z± 1 = sec.', cot 1 = cosec.", and the reduction from the proportional to the divisional form of the other five tan. = l sec. = 1 , cot. =cos. = 1 tan.= cos. cos. sin, sin.' 1 ; the various functions being expressed in terms of the as cot.
sumed unit. Thus, in the right-angled triangle ABC 2), if AC be radius, BC = sin., and AB = cos., of the angle A; but if the radius be assumed as unity, sin. A = - ``, cos. A = and similarly from the above tan. = B, see. = LB, and = j? ; and i cot. = n algebraic trigonometry these latter are the defini tions of the trigonometrical functions.
The only angular functions which geometry enables us to determine with accuracy are those belonging to the angles of an equilateral triangle 1), no isosceles right angled triangle (Euc. II. 9), and an isosceles triangle which has cacti of the angles at its base double of the third angle (i.e., base angles each vertical angle, 3v') ttuc. IV_ 10); and from these, by means of a proposition (demonstrated iu all text books on the which determines the functions of the angle -}- B) from a knowledge of tire functions of A and of those of B; and also, as a corollary to the the functions of2A, 4A, 8A, and inversely of A, +A, etc., from a knowledge of those of the angle A, have been obtained and tabulated the functions of all angles from 1' to 45, the functions of angles from 4.5° to 360° being, as is evident from the above remarks respect ing complementary and supplementary angles, merely repetitions of these.
The relations between the angles and sides of a triangle are three in number, and :are obtained front simple geometric they are AB : AC : sin. C : sin.
AB i }- — ACa B; (2) cos. B = 2XB. BC ; AB -}- AC :AB—AC: : tau. 1(B + C) : tan. (B — C). From these relations in conjunction with the fact that the three angles of a triangle collectively amount to 180°, it is having give any Three (one being always a of the six elements sides and three of a to deter mine time other three. It is this that constitues trigonometry in its primitive and elemen tary form. If the triangles be only the first relation and the property of tliu sides of a right-angled triangle, are, necessary for the complete solution. Further info rniation on this subject will be found in any textbook.
Algebraic trigonometry is one of the most important branches of but it is too extensive and varied to be even sketched here; suffice it to say, that in it the trigo noinetrical functions are not considered as geometrical magnitudes, but as numerical quantities having certain relations to each other, and that the circle as well as the angu lar functions are treated as multiples or of the radius. Many important results, such as the approximate estimation of the circumference of a circle, the contple tiou of the solution of cut is etc., have been obtained its means: and a, thorough knowledge of its modes and results is absolutely necessary to an acquaintance with higher mathematics.
Spherical trigonometry is plane trigonometry applied to spherical See any textbook.
or TuiKALA, at. of European Turkey, 33 m. w.s.w. of Larissa. It is built on the slope of chill, manufactures cotton and woolen and hasa large transit trade with Epirus and Albania, The neighboring plains which are watered by the Salem bria (sna.Peneus), are rich in all sorts of fruits. Pop. about Triklala is the Trikkm of Homer, and was celebrated in the classic ages for its temple of sculapius.
, a small natural order of plants, belonging to the elas dietyogens (q.v.) of They are Herbaceous plants, with tubers or rootstocks, whorled leaves, hermaphodrite perianth of six leaves, the three inner leaves sometimes colored: six, or ten the filaments extending beyond the anthers in awl shaped the ovary free, 3 to 5-celled, with numerous the fruit succnleut. The order is characterized by narcotic The genus Paris belongs to it.