Angles unlimited in magnitude. Now OP may start revolving from OX, make one complete revolution, continue to revolve, and then cease revolving when it has again reached the position

Terminal Line

P

Initial Line

X

Or,

OQ. This is indicated in Fig. 33. OP may make two complete revolutions. before it comes to rest in the position OQ; or, it may make three revolutions, or four, or as many as one please, before ceasing its revolution at the position OQ. An angle of 360° is described each time that OP makes a complete revolution, and OP can make as many revolutions as one please. According to the trigonometric definition of an angle, therefore, angles are unlimited in magnitude.

FIG. 33.

Terminal Line

Moreover, when this definition of an angle is adopted, the same figure can represent an infinite number of different angles. Any two of these angles differ from each other by a whole number of complete revolutions. For instance, Fig 34 may represent 60°, 360°+60° or 420°, 2.360° + 60° or 780°, 3·360° +60° or 1140°, ..., n. 360° + 60°, in which n denotes any whole number. Any two of these angles differ by a multiple of 360°. Angles which have the same initial and terminal lines may be called coterminal angles.

X

Initial Line

FIG. 34.

Positive and negative angles. The revolving line OP (Fig. 32) may revolve about O in the same direction as that in which the hands of a watch revolve, or it may revolve in the opposite direction. The following convention (see Art. 36) has been adopted for the sake of distinguishing these two opposite directions:

When the turning line revolves in a counter-clockwise direction, the angles described are said to be positive, and are given the plus sign; when the turning line revolves in a clockwise direction, the angles described are said to be negative, and are given the minus sign.

Thus, for example, Fig. 34 represents the angles + 60°, — 300°; further, this figure represents the angles 60° ±n 360°, in which n denotes any whole number. The angle - 300° is included in these angles, for, on putting -1 fcr n, there is obtained 60°-360°, i.e.300°. (Negative angles are also unlimited in magnitude.)

38.]

SUPPLEMENT AND COMPLEMENT.

69

As in the case of lines, the sign of an angle can be denoted by the order of the letters used in naming the angle. Thus XOQ denotes the angle formed by revolving OX toward OQ, and QOX denotes the angle formed by revolving OQ toward OX. Accordingly, QOX-XOQ.

Quadrants. In Fig. 32, XOY, YOX1, X1OY, YOX, are called the first, second, third, and fourth quadrants, respectively. When the turning line ceases its revolution at some position between OX and OY, the angle described is said to be an angle in the first quadrant; when the final position of the turning line is between OY and OX1, the angle described is said to be in the second quadrant; and so on for the third and fourth quadrants.

For example, the angles 30°, 345°, 395°, 725° are all in the first quadrant; the angles — 60°, 340°, 710° are all in the fourth quadrant; the angle - 225° is in the second quadrant, and the angle 225° is in the third quadrant. NOTE. While all acute angles are in the first quadrant, not all angles which are in the first quadrant are acute.

EXAMPLES.

NOTE. When it is necessary, the number of revolutions and their direction may be indicated on the figure in the manner shown in Fig. 34.

Lay off the following angles with the protractor: In the case of each angle name the least positive angle that has the same terminal line. Name the quadrants in which the angles are situated. In the case of each angle name the four smallest positive angles that have the same terminal line.

1. 137°, 785°, 321°, 930°, 840°, 1060°, 1720°, 543°, 3657°.

2. - 240°, 337°, 967°,

830°, — 750°, — 1050°,

7283°.

3. − 47° + 230° + 37°, 420° — 470° + 210° — 150°, 230° – 47° + 37°, 230° + 37° 47°.

38. Supplement and complement of an angle. The supplement of an angle is that angle which must be added to it in order to make two right angles, or 180°; the complement of an angle is that angle which must be added to it in order to make one right angle, or 90°. Thus, if A be any angle, then

supplement of angle A= 180° — A,

complement of angle A= 90° - A

EXAMPLES.

1. What are the complements and supplements of 40°, 227°,

2. By means of a figure verify the results obtained in Ex. 1.

3. What are the complements of — 295°, — 324°, 200°, 240°, - 110°,

230°, 150°, - 40°, 340°, 75°, 83°, 12°,

167° ?

4. What are the supplements of the angles in Ex. 3?

5. Verify the results in (3), (4), by drawing figures.

39. The convention of signs on a plane. Articles 36, 37 contain statements of the conventions adopted regarding the algebraic

FIG. 35.

P

X

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 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:

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.

The distance of points, such as P1, P2, P3, P4, from XX, is always measured from X,X toward the points.

Any turning line (or oblique line) as OP is measured positively

40.]

GENERAL DEFINITION OF RATIOS.

71

from 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 X,OX from the point O to the foot of the perpendicular MP, and MP is the distance from XOX to the point P. Following are the definitions of the trigo nometric 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 positive or negative, as indicated in the following table: