Conventions and Definitions.— Any two perpendicular lines, as X'OX and YOY', divide the unbounded plane into four congruent por tions, called quadrants, which hang together about the origin 0 and of which XOY . YOX', X'OY' and Y'OX, taken in order, are known respectively as the 1st, 2d, 3d and 4th.
The amount .of turning (about 0 and in the plane) of a half-line, as OX, which will bring 'it to coincide with another half-line, as OP, is called the angle between them. The angle is regarded positive or negative according as the turning is counter-clockwise or clockwise. If a half-line turn quite round, so as to coincide with its initial position, it is said to turn through or to generate a whole, or round, angle, or pen gon. The unit-angle used in practical com putation is the 360th part of the whole angle, and is named degree. The unit-angle employed in theoretic investigation is the angle generated by the turning of a half-line, as OP, till one of its points, as P, describes a circle arc equal in length to the segment OP. This unit-angle is named radian, being subtended by an arc equal to its radius. Either unit is readily expressible in terms of the other. Thus the whole angle, or 360°, being equal to 2ar radians, where a is the ratio of the circumference to the diameter of a circle (see GEOMETRY, ELE MENTARY; GEOMETRY, PURE PROJECTIVE) , it is seen that 1 radian = 57.3° (approximately), ir radians = 180°, ff = 90° 7r = etc. From the definition of angle, it is clear that an angle may exceed Suppose, for example, that OX generates an angle a (Fig. 1) and then turns through br; it will thus have generated the angle a + 27r. The half-lines bounding an angle are called its beginning and end. Except when otherwise stated or unmistakably implied, all angles will be thought as beginning at OX, origin of angles.
X'OX and YOY' are named co-ordinate axes (see GEOMETRY, CARTESIAN) ; the former, the axis of abscissa, or X-axis; the latter, the axis of ordinates, or Y-axis. Distances on or parallel to the X-axis are considered positive if measured rightward negative if leftward; dis tances on or parallel to the Y-axis are positive if measured upward, negative if downward. Accordingly, to each point in the plane (Fig. 1) corresponds a unique pair of numbers, its abscissa and ordinate, its x and y, and con versely. ' Definitions of the Trigonometric Func On OP, making any angle a with OX, take any point P. Complete the figure as indi sated, denoting the lengths, of OF, FP and OP by x, y and r respectively. Plainly the values of x, y and r vary with the position of P on OP, but, and this is important, by the laws of similar triangles, their ratios do not. These
ratios do vary, however, and this, too, is im portant to note, with the value or size of the angle a. Because of this reciprocal dependence of the ratios on a and of a on the ratios, the latter are called functions of a, and conversely. Because of their importance, the ratios, or trigonometric functions of the angle have re ceived names and symbols, as follows: —sine of a —sin a, — ...cosecant of a —cosec a, y x — —cosine of a cos a, r x — —secant of a —sec a, —tangent of a —tan a, —cotangent of a —cot a.
7 These equations serve to define sin a and cos a for all finite values of a. For any one of the other functions the denominator of the defining ratio becomes zero for certain values of a. Division by zero, being meaningless, is inadmissible. For such values of a, therefore, these functions are not defined. Thus, e.g., the tan is not defined for a = 90°. It is indeed cus tomary to write tan 90" = co, but this is merely short for saying that, as a approaches 90°, tan a increases beyond every assignable (finite) value.
Range of Variation and It is readily seen that, if a (Fig. 1) increase or de crease by 27r, then x y and r regain their initial values. Hence sin (a 2nir) = sin a, n being any positive or negative integer. Similarly for the other functions. Hence the functions are periodic, having the period 27r. It is to be noted, however, that an increase or decrease of a by it alone merely reverses x and y in sign, their ratios remaining the same. Accordingly the period of tan and cot is 7r. The periodicity of the trigonometric tunctions, it is, that gives them their great value in Analysis. Each func tion assumes all the values of its range as a varies through the period of the function. As a varies continuously from 0 to 27r, sin a changes continuously, increasing from 0 for a = 0 to 1 for a = ' then decreasing to 0 for a.= a, then changing sign and decreasing (alge braically) to —1 for a= — then increasing to 0 for a.= 2r. Meanwhile cos a runs ously through the same circuit of values, though in different order, beginning and ending with 1, IT and changing sign at a =— 2 and 2 — • Hence sin and cos are restricted to the values 1,— 1 and intermediate values. Not so the other func tions, however. Both tan and cot assume all finite real values, and also vary continuously with a except for those angle values for which, as noted, the definitions fail. For any such value, the function is said to be discontinuous. For example, as a increases through 90°, tan a leaps from being great at will and positive to being great at will and negative. The sec and the cosec each assumes every finite real value except those between 1 and —1.