TRIGONOMETRY (from Gk. trigonon, triangle, from rpe7s, treis, three ± 7copla, !Ionia, angle + -tarpla, metria, measure ment, from pop, at et ron, measure, from ilerper, 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.
functions are tangent of O. symbolically written a tan8=w;cotangentof0.orcoto= a secant of 0, or sec 9 = ; and cosecant of 0 or cosec 0 There are also used the functions vcr a sine of 0, or vers 9 = 1 ; and coeersine of 9, or covers a =1 . The propriety of calling these ratios 'functions' of the angle 9 consists in this, that the value of any ratio depends upon the value of That is, in any right-angled triangle All'C', having an acute angle 0, the b' corresponding ratios . -7 (Fig. 2) are equal to the ratios c a c . b ' , and in any angled triangle in which the acute angle is not equal to 0 the corresponding ratios are not equal to those for O. The trigonometric func tions as defined by the above ratios are evi dently limited to angles less than 00°, 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.
The common functions of trigonometry may he defined as ratios of certain sides of a right tri angle. Thus, in the figure, the ratio is called the sine of the angle 0, commonly written sin 0 a The ratio is called the cosine of c the angle 0, written cos 0= 7. The other In Fig. 3 the radius OA (= OB) may be re garded as the unit of length, hence the ratio 13,1?I = 13,111, and sin A0B, = B,D[, Similarly cosA0B, = OM, tanAOB, = AT, cotA0B, = PQ, seeA(111, = OT, cosecAOB, = OP, versA011, MA, and coversA013, = CP,. If the angle is
obtuse as or reflex as A0B,, the functions arc represented by the corresponding lines. E.g. sin AOB, = tan A011, = AT'. The following convention of signs (see Fig. 4), however, serves to associate these values with the proper angle: Lines measured to the right of the vertical diameter, as OM, are called positive, and those to the left, as OM negative; lines measured upward, as II M, from the horizontal diameter are called positive, and those downward, as negative; the revolving radius OB, is always positive. Thus (see Fig. 4) the signs of the functions of an angle not exceeding 90° are all plus. The versine and coversiue are evidently always positive. The sine and thy cosecant are positive in the first and second quadrants, and negative in the third and fourth. The cosine and the secant are positive in the first and fourth quadrants and negative in the second and third. The tangent and cotangent are positive in the first and third quadrants, and negative in the second and fourth. We have, therefore, the following relations: press cos(270° 0) in terms of a function of P, This angle is in the third quadrant and therefore its cosine is negative. To make the angle 270° 0 less than 90' we must subtract 180° and we have cos(270° 0) = cos(90° 0). But cos(90° = sin , and we have = sin P.
The increasing of an angle by 360° or any mul tiple of 360° does not alter the value of the trig onometric functions of that angle. (See FUNC TION.) It appears from the geometric representa tion of the functions that the values of the sines and cosines of all real angles lie within the inter val -)- 1, 3: the values of the tangents and co tangents of all real angles lie within the interval cn and m ; those of the secants and co secants without the interval + 3, 1, as is shown in the following table: The variations of the functions are best ex hibited by means of graphs. In the figures the arcs are laid off as abscissas and the functions as ordinates. See COURDINATES.