Figs. 59 and 74, approach more and more nearly squares as the number of intervals in the circle is increased. Each of the ratios in (2) approaches as near as we please to unity the smaller x is taken, but the limits of these ratios are unity only when the angles are measured in radians. The word "limit" used above stands for the same concept that arises in elementary geometry. It may be formally defined as follows: DEFINITION: A constant, a, is called the limit of a variable, t, if, as t runs through a sequence of numbers, the difference (at) becomes, and remains, numerically smaller than any preassigned number. 81. Graph of cot x. In order to lay off a sequence of values of cot on a scale, it is convenient to keep the denominator constant in the ratio (abscissa)/(ordinate) which defines the cotangent. P11 P10 P9 P8 P7 P5 P4 P3 P2 Pi D11 D5 D4 Dз D2 D1 D10 D9 D8 DT The denominator may also, for convenience, be taken equal to unity. Thus, in Fig. 75, the triangles of reference D1OP1, D2OP 2, for the various values of @ shown, have been drawn so that the ordinates P1D1, P2D2, are equal. If the constant ordinate be also the unit of measure, then the sequence OD1, OD2, OD 3, ODT, OD8, represents, in magnitude and sign, the cotangents of the various values of the argument 0. Using OD1, OD2, as the successive ordinates and the circular measure of O as the successive abscissas, the graph of y cot x is drawn, as shown by the dotted curve in Fig. 74. = The sequence OD1, OD2, Fig. 75 is exactly the same as the sequence OT, OT 2, Fig. 74, but arranged in the reverse order. Hence, the graph of the cotangent and of the tangent are alike in general form, but one curve descends as the other ascends, so that the position, in the plane xy, of the branches of the curve are quite different. In fact, if the curve of the tangents be rotated about OY as axis and then translated to the right the distance π/2, the curves would become identical. Therefore, for all values of x: BN' FIG. 76.-Graphical Construction of y = sec x. 82. Graph of y = sec x. Since sec is the ratio of the radius divided by the abscissa of any point on the terminal side of the angle 0, it is desirable, in laying off a scale of a sequence of values of sec 0, to draw a series of triangles of reference with the abscissas in all cases the same, as shown in Fig. 76. In this figure the angles were laid off from CQ as initial line. Thus: or, if CS, be unity, the distances like CTs, laid off on CQ, are the secants of the angles laid off on the arc QS50 or laid off on the axis OX. The student may describe the manner in which the rectangles made by drawing horizontal lines through the points of division on CQ and the vertical lines drawn at equal intervals along OX, may be used to construct the curve. If the radius of the circle be 1.15 inches, what should be the length of Or in inches? The student may construct and discuss the locus of y = CSC x. Compare with the locus y = sec x Exercises 1. Discuss from the diagrams, 59, 74, 76, the following statements: Any number, however large or small, is the tangent of some angle. The sine or cosine of any angle cannot exceed 1 in numerical value. The secant or cosecant of any angle is always numerically greater than 1 (or at least equal to 1). 4. Describe fully the following, locating nodes, troughs, crests, asymptotes, etc.: = 83. Increasing and Decreasing Functions. The meanings of these terms have been explained in §26. Applying these terms to the circular functions, we may say that y sin x, y = tan x, y = secx are increasing functions for 0 < x <π/2. The cofunctions, y = cos x, y cot x, y = csc x, are decreasing functions within the same interval. = Exercises Discuss the following topics from a consideration of the graphs of the functions: 1. In which quadrants is the sine an increasing function of the angle? In which a decreasing function? 2. In which quadrants is the tangent an increasing, and in which a decreasing, function of its variable? 3. In which quadrants are the cos 0, cot 0, sec 0, csc 0, increasing and in which are they decreasing functions of ? 4. Show that all the co-functions of angles of the first quadrant are decreasing functions. FIG. 77.-Construction of the Rectangular Hyperbola. 84. The Rectangular Hyperbola. We have seen that the circle is the locus of a point whose abscissa is a cos 0 and whose ordinate is a sin 0. The rectangular, or equilateral, hyperbola may be defined to be the locus of a point whose abscissa is a sec 0 and whose ordinate is a tan 0. To construct the curve, divide the X-axis proportionally to sec 0, and the Y-axis proportionally to tan 0, as shown in Fig. 77. The scale OX of this diagram may be taken from OY of Fig. 76, and the scale OY may be taken from OY of Fig. 74. The plane of xy may be divided into a large number of rectangles by passing lines through the points of division perpendicular to the scales and then, starting from A and A', sketching the diagonals of the successive cornering rectangles. The parametric equations of the curve are, by definition: The Cartesian equation is easily found by squaring each of the equations and subtracting the second from the first, thus eliminating by the relation sec2 0 — tan2 0 = 1: This is the Cartesian equation of the rectangular hyperbola. The equation of the rectangular hyperbola may also be written in the useful form: Compare (1) and (3) with the equations of the circle. = (3) The rectangular hyperbola here defined will be shown, in §86, to be the curve 2xy a2 rotated 45° clockwise about the origin. 85. The Asymptotes. Let G'G be the line y slope of OP1 is PD /OD or y /x or = x, Fig. 77. The The value of corresponding to the point P is AOH. As the point P moves upward and to the right on the curve, the angle e, or AOH, approaches 90° and sin approaches unity. Hence the line OP approaches OG as a limit, and P approaches as near as we please to OG. The same reasoning applies to points moving out on the curve in the other quadrants. The lines GG' and JJ' are called asymptotes to the hyperbola. = a2 and x2 y2 = a2. In Fig. 78, let = 1 86. The Curves 2xy the curve be the locus 2x11 a2, referred to the axes X'X1 and Yi'Y1. This curve has already been called the rectangular hyperbola. (See §23.) We desire to find the equation of the curve 1 To avoid an excessive number of construction lines, OP is not shown in the figure. |