ZBOC equal to 30°, and make CD = 3r, where r is the radius. Then AD is approximately equal to the semicircumference πг. Find to three decimal places the approximate value of π given by this construction. 58. Angles of any Magnitude. Let OX and OP be two lines drawn from an initial point 0. Let a line start from co Ө FIG. 82. X incidence with OX, the initial line, and rotate about 0, coming to rest finally in coincidence with OP, the terminal line. The line may rotate in either direction, and it may make any number of revolutions before coming to rest. The line is said to generate an angle whose mag nitude is determined by the amount and direction of the rotation. The numerical value of the magnitude may be given in degrees, right angles, or revolutions. The sign is positive or negative according as the direction of rotation is counter-clockwise or clockwise, i.e., in the opposite or in the same direction as the hands of a clock rotate. If the terminal line of a first angle is the initial line of a second the sum of the angles is defined to be the angle whose initial line is that of the first, and whose terminal line is that of the second angle. This is analogous to the sum of two lines (page 13). •210° y 300° FIG. 83. Two lines determine a countless number of angles. If is any one of them, the others differ from 0 by an integral multiple of 360°. They may all be represented by 0 + n360°, where n = ± 1, ± 2, ± 3, 3,. X The arcs in Fig. 82 indicate the three angles: 0 = 225°, 0' = 0 135°, 0′′ = 0 + 360° = 585°. If the initial line of an angle coincides with the positive part of the x-axis, the angle is said to lie in the quadrant in which the terminal line lies. Thus in Fig. 83, the angle 300° lies in the fourth quadrant, -210° in the second. The positive direction on the terminal line in such a figure is defined to be away from the origin. For example, the positive direction on the terminal line of an angle of 180° is to the left. We shall make an important use of the angles whose terminal lines bound, bisect, or trisect the four quadrants. 59. Trigonometric Functions of any Angle. Let be the number of degrees in any angle whose initial line coincides with the positive part of the x-axis, let P(x, y) be any point on the terminal line, and let OP = r. Consider the ratio y/x, which is a negative number for the case indicated in the figure, since x = OM is negative and y = MP is positive. The numerical value of y/x may be found approximately by measuring УА мхо FIG. 84. A8 the lengths of MP and OM and dividing the former by the latter. P P If P'(x', y') is any other point on the terminal line, the ratios y'/x' and y/x have the same sign, and also the same numerical value, since the triangles OMP and O'M'P' are similar. Hence if is given a definite value, the value of y/x is determined, and therefore the ratio y/x is a function of 0. Similarly, the ratio of any one of the numbers x, y, r, to any other is a function of the angle 0. These functions are called trigonometric functions. They are named in accordance with the definitions below, which hold for Fig. 86 A, B, C, D. M' M FIG. 85. DEFINITIONS. If is any angle whose initial line coincides with the positive part of the x-axis, if P(x, y) is any point on the terminal line, and if OP r, then Since the three ratios on the right are the reciprocals of those on the left, we have the reciprocal relations: The first table in Section 57 gives the sine, cosine, and tangent of the angles 30°, 45°, and 60°. The values of the sines may be easily remembered by noticing that they are respectively {√1, 1√2, 1√3, while the cosines are the same numbers in the reverse order. The tangent of any one of the angles may be obtained by dividing the sine by the cosine. For we always have y/r sin 0 x/r cos tan 0 X (2) If is in quadrant I, x, y, and r are positive, and the values of the six ratios, or functions, are positive. But if 0 is in one of the other quadrants, either x or y, or both, are negative, although r is always positive, and hence these ratios may be negative. For example, if 0 is in the second quadrant, tan 0 is a negative number, since x is negative and y is positive. y/x The reciprocal relations show that the signs of cot 0, sec 0, csc 0, for a given value of 0, agree respectively with the signs of tan 0, cos 0, sin 0. Hence it is necessary to fix in mind the signs of the latter functions only. This is readily done in connection with the graphs (see Section 61). If the terminal line of bounds, bisects, or trisects one of the quadrants, the functions of ◊ may be found directly from the definitions, as in the examples following, by methods of elementary geometry. EXAMPLE 1. Find the functions of 0°. Let P(x, y) be any point on the terminal line, which coincides with the positive part of the x-axis, since 0 = 0. Then not exist. However, cot is the reciprocal of tan 0, by (1). As 0 ap proaches zero, tan approaches zero, and hence cot becomes infinite as 0 approaches zero (IV, page 118). This is sometimes indicated by the symbols cot 0° ∞ In like manner, as 0 approaches zero, csc @ becomes infinite. The acute angles of the triangles OPM are 30° and 60°. Hence the numerical value of OP is twice that of MP. If we let the numerical value of MP be 1, then that of OP is 2, and hence that of OP is 3. But the coördinates of P are negative. Hence we may take + tan 210° y/x sin 210° = y/r csc 210° = r/y = 2/(− 1) = − 2. cos 210° = x/r = −√√3/2. sec 210° = r/x = 2/(−√/3) = − 2√3/3. − 1/(− √/3) = √√/3/3. cot 210° = x/y = − √3/(−1) = √3. Since the functions of two angles with the same terminal line may be defined by means of the same triangle, and since all angles with the same terminal line may be expressed in the form 0+ n360° (Section 58), we where n FIG. 89. n = ± 1, ± 2, ± 3, etc. DEFINITION. A function is said to be periodic if its value is unchanged when the value of the variable is increased by a constant, that is, if f(x + c) = f(x). If c is the smallest constant of this sort, it is called the period of the function. For example, the height of the tide at the seashore is a periodic function of the time. For if approximately 12 hours and 25 minutes are added to the time, the height of the tide will be the same. |