37. Trigonometric definition of an angle. Angles unlimited in magnitude. Positive and negative angles. In books on plane geometry a plane angle is defined in various ways, namely, as the inclination of two lines to one another, which meet together, but are not in the same direction; or, as the figure formed by two straight lines drawn from the same point; or, as the amount of divergence of two lines which meet in a point, or would meet if produced; or, as the opening between two straight lines which meet; or, as the difference in direction of two lines which meet; and so on. In these definitions an angle is always regarded as less than two right angles. A definition according to which angles are less restricted, is adopted in trigonometry.

Y

P

Trigonometric definition of an angle. The angle between two lines which intersect is the amount of turning which a line revolving about their point of intersection makes, when it begins its revolution at the position of one of the two lines and stops in the position of the other line. Thus, for example, the angle between OX and OQ is the amount of turning which is made by a line OP revolving about O when OP starts revolving from the position OX and stops its revolution at the position OQ. The line OX at which the revolution begins, is called the initial line; the line OQ at which the revolution ends, is called the terminal line; when the turning line OP has reached the terminal position OQ, OP is said to have described the angle XOQ.

QTerminal Line

Initial Line

X

FIG. 32.

Let YOY1 be at right angles to X1OX. When OP has revolved until it lies in the position OY, it has described a right angle, or 90°; when it has revolved until it lies in the position OX,, it has described two right angles, or 180° (this is usually termed "a · straight angle" or "a flat angle"); when OP keeps on turning until it is in the position OY1, it has described three right angles, or 270°; when OP has again reached the position OX, that is, when it has made one complete revolution, it has described four right angles, or 360°.

Terminal Line

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 OQ. This is indicated in Fig. 33. Or, 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.

Initial Line

FIG. 33.

X

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 for n, there is obtained 60°-360°, i.e. 300°. (Negative angles are also unlimited in magnitude.)

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, all angles which are in the first quadrant are not 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 — 230°, 150°, – 295°, — 324°, 200°, 240°, — 110°, - 167° ?

· 40°, 340°, 75°, 83°, 12°,

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

X1

姫

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 O, and YOY1 is at right angles to XOX. The following 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 X1X, 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 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 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.]