EXAMPLES. 1. What are the complements and supplements of 40°, 227°, — 40°? supplement of 40° = 180° 2. By means of a figure verify the results obtained in Ex. 1. 3. What are the complements of - 230°, 150°, — 40°, 340°, 75°, 83°, 12o, 295°, - 3240, 200°, 240°, - 110°, — 167° ? 4. What are the supplements of the angles in Ex. 3? 5. Verify the results in (3), (4), by drawing figures. P 39. The convention of signs on a plane. Articles 36, 37 contain statements of the conventions adopted regarding the algebraic signs to be given to distances measured on parallel straight lines, and to angles described by the revolution of a turning line. A figure, such as Figs. 32, 35, will be frequently used in the X articles that follow. In this figure, OX is the initial line, the turning line revolves about 0, and YOY, is at right angles to XOX. The fol lowing convention has been adopted regarding the lines which will be used: FIG. 35. Horizontal lines measured in the direction of X are taken positively; Horizontal lines measured in the direction of X1 are taken negatively; Vertical lines measured upward are taken positively; Vertical lines measured downward are taken negatively. 29 The distance of points, such as P1, P2, P3, P4, from XX, is always measured from XX toward the points. Any turning line (or oblique line) as OP is measured positively from O toward the end of the turning line which lies in the direction of X from O when the turning line coincides with the initial line. Thus a distance + 3 on OP will terminate at T, distant 3 units from O, and a distance 3 on OP will terminate at T1, distant 3 units from O, but in the direction opposite to the former. This is sometimes briefly expressed in the words: the turning line carries its positive direction with it in its revolution. 40. General definition of the trigonometric ratios. The remarks in this article apply to each of the four figures below. In each figure, O is the point about which the angle is described, OX is the initial line, and OP is the terminal line. The first figure represents any angle in the first quadrant; the second figure represents any angle in the second quadrant; the third figure, any angle in the third quadrant; and the fourth figure, any angle in the fourth quadrant. In each figure the angle will be called A. Let P be any point in OP, the terminal line of any angle A. From P draw PM at right angles to the initial line OX, or to the initial line produced in the negative direction. In each figure, OM is the distance measured along X1OX from the point 0 to the foot of the perpendicular MP, and MP is the distance from X10X to the point P. Following are the definitions of the trigonometric ratios; these definitions apply to the angles represented in Fig. 36, and, accordingly, to all angles whatsoever. [Particular attention should be paid to the order of the letters used in naming the lines, for this order indicates the direction in which the line is measured. See Art. 36.] OP The ratio is called the secant of the angle A. OM Inspection will show that the definitions of the trigonometric ratios for acute angles given in Art. 12, are in accordance with these general definitions. N.B. The projection definitions of the trigonometric ratios are given in Note B, Appendix. 41. The algebraic signs of the trigonometric ratios for angles in the different quadrants. Figures 36 show that if the angle A is in the first, second, third, fourth quadrants, then the algebraic sign of MP is +, +, −, −, respectively, and the algebraic sign of OM is +, -, -, +, respectively. As stated in Art. 39, OP is always taken positively. Hence, on paying regard to the algebraic signs of OM, MP, OP, in the several quadrants, it will be seen that the ratios of the angles in these quadrants are posi tive or negative, as indicated in the following table: The student is advised to preserve his work on these examples. If he regards his results attentively, he will probably discover some useful facts, and be able to deduce some useful theorems, concerning angles in general. Any preceding results, such as those in Art. 15, may be used as an aid in solving these exercises. 1. Find the ratios of 945°. 945° 2 × 360° + 225°. X1 M-1 0 X را FIG. 37. .. OP, the terminal line of angle 945°, has the position shown in Fig. 37. For this position of the terminal line, OM and MP are both negative. As shown in Art. 15, the lines OM, MP, OP, in this figure are respectively proportional to 1, 1, √2. These are indicated on the figure with their proper algebraic signs. It is immediately apparent that 2. Construct, and find the ratios of, 420°, 780°, 1140°. 3. Construct, and find the ratios of, 120°, 480°, 240°, 600°, — 60°, 300°, 660°, - 720°. 4. Construct, and find the ratios of, 150°, 410°, 210°, −150°, 330°, −390°. 5. Construct, and find the ratios of, 45°, 765°, 135°, — 225°, 225°, 585°, - 405°, 1035°. 6. Construct, and find the ratios of, — 754°, 487°, — 245°. 135°, 7. Compare the ratios of 90° - 30°, 90° – 60°, 90° — 45°, 90°. 90° – 240°, 90° – 300°, with the ratios of 30°, 60°, 45°, 135°, 240°, 300°, respectively. 8. Compare the ratios of 90° + 30°, 90° + 60°, 90° + 45°, 90° + 135°, 90° +240°, 90° +300°, with the ratios of 30°, 60°, 45°, 135°, 240°, 300°, respectively. 9. Compare the ratios of 180° - 30°, 180° - 60°, 180° - 45°, 180° — 135°, 180° — 240°, 180° – 300°, with the ratios of 30°, 60°, 45°, 135°, 240°, 300°, respectively. So, also, the ratios of — 30°, — 60°, - 45°, etc. 10. Are any general relations indicated by the results of Exs. 7, 8, 9? If so, state these relations. Try to prove them. 42. To represent the angles geometrically when the ratios are given. In constructing the angles in this article it is necessary to bear in mind that, according to the definitions given in Art. 40: When MP is positive, it can be drawn in the first and second quadrants; EXAMPLES. P P 1. Represent by a figure the angles which have sines equal to . Calculate their other ratios. Let A denote an angle whose sine is ; i.e. let MP sin A. But sin A = (Art. 40). Hence, MP:OP=3:4; and if OP OP= 4, then MP = 3. Now, MP can be drawn positively in both the first and second quadrants. Hence the problem amounts to finding a point in the first quadrant and a point in the second quadrant, each at a distance 4 IY from O and a distance 3 from X10X. The result is indicated in Fig. 38. The student can make the construction for himself. The angles having sines equal to 4, accordingly, include all the angles which have OP for a terminal line, and all the angles which have OP1 for a terminal line. By Art. 37 each of these two sets of angles consists of an infinite number of angles, any two of which differ from one another by a whole number, positive or negative, of complete Y FIG. 38. X revolutions. A general algebraic expression which includes all these angles, is deduced in Art. 85. |