TRIGONOMETRY (Gr. trigonon, a triangle, metria, measurement), the measurement of triangles. This definition, though expressing correctly enough the scope of trigo nometry in its early stages, is now wholly inapplicable, as trigonometry, like geometry, has far exceeded its primitive limits; and though the original name is, for convenience, retained, the science may be more properly defined as the "consideration of alternating or periodic magnitude." Trigonometry, within the limits of its earlier definition, is geo metrical; its advance beyond theselimits is due to the introduction of purely algebraic methods. The quantities with which geometrical trigonometry has to deal are certain lines definitely placed with respect to an angle, and consequently varying with it. These lines, generally denominated trigonometrical functions of the angle, arc the sine, cosine, tangent, cotangent, secant, and cosecant, and are represented in the accompanying figure. The angle BAC is placed at the center of a circle, called the circle of reference; its sine, `CD, is the perpendicular let fall from the extremity of one radius upon the other; the cosine, DA, is that part of the radius be tween the foot of the sine and the center; the ltengent, BE, is drawn at right angles to one radius to meet the other produced: the secant, AE, is the radius produced to meet the extremity of the tangent; the cotangent, FG, is drawn from the extrem ity of a radius at right angles to one of the former, to meet the other produced: and the cosecant, AG, is the radius produced to meet the extremity of the cotangent. Oth er functions, as the versed sine, DB, which is the distance from B to the foot of the sine, and its counterpart, the coversed sine, FIT, have been occasionally introduced and defined, but they are of no practical use. EAF, the angle which must be added to BAC to make up a right angle, is called the complement of BAC; and CAL, the defect of BAC from two right angles, is called its supplement; and by inspection of the figure we can see at once that the sine of BAC, CD, is equal to All, the cosine of its comple ment; that the cosine of BAC, AD, is equal to CII, the sine of its complement; and that generally any function of an angle is the co-function of its complement, and rice rel.* also, that CD, the sine of CAB, is also the sine of its supplement; AD, the cosine of CAB, is the cosine of its supplement; and that generally the function of an angle is the function of its supplement. If a right angle be added to BAC, then we have the tri angles ADC, ABE, shifted so as to be situated in the same relative position to AF as they now are to AB, and each line is consequently at right angles to its former position; hence the sign of BAC is the cosine of (90° ± BAC), and similarly of the others. By an
extension of this process of investigation we arrive at the general conclusions that if an angle be added to or taken from one or an odd number of right angles, the function of the original angle is the co function of the one so derived; and that if an angle be added to or taken from an even number of right angles, the functions of the original angle are the func tions of the derived one. But since a function of an angle is the same function of its supplement, a knowledge of the function would not enable us to determine to which of the two angles it unless we possessed some knowledge of more than the mere magnitude of the function. This desideratum is supplied in the following manner: B is taken as the zero-point of reckoning, the radius BA, which is thus supposed to be fixed, is one of the bounding lines of every angle, the other side being supposed to move in the direction BFL, as the angle increases. Let the radius AC be supposed to sweep round the circle in a left-hand direction (viz., toward F), then, as it approaches F, the sine CD increases, till, on reaching F, the sine coincides with the radius: passing F, and moving toward L, the sine diminishes, till, on reaching L, it becomes zero. Con tinuing its progress round the circle, the angle BAC becomes re-entrant (viz., greater than two right angles); and its sine again increases, becoming equal to the radius at 31, and diminishing in the fourth quadrant till it becomes zero at B. While the angle increased from B to L, the sine was drawn downward; for the other half of the revolu tion, it was drawn upward; hence, in the first and second quadrants, the sine is said to be positive, and in the third and fourth, negative, the position of a function in the first quad rant being adopted as the standard. The following table shows the variation (increase or decrease, and between what limits, as well as the sign affecting it) of each of the func tions as the angle increases: We here observe that all the functions increase and decrease alternately as the angle of which they are the functions passes from one quadrant to another; also that the sine and cosecant are affected by the same signs, as also are the cosine and secant, and tan gent and cotangent.