CHAPTER V.

TRIGONOMETRIC RATIOS OF ANGLES IN GENERAL.

36. Directed lines.

Let MN be a line unlimited in length in the directions of both M and N. Suppose that a point starts at P and moves along this line for some given distance. In order to mark where the point stops, it is necessary to know, not only this distance, but also the direction in which the point has moved from P. This direction may be indicated in various ways; by saying, for instance, that the point moves toward the right from P, or toward the left from P; that the point moves toward N, or toward M; that the point moves in the direction of N, or in the direction of M; and so on. Mathematicians, engineers, and others

have agreed to use a particular method (and this practically comes to the adoption of a particular rule) for indicating the two opposite directions in which a point can move along a line, or in which distances along a line can be measured. This convention, or rule which has been adopted for the sake of convenience, is as follows:

Distances measured along a line, or along parallel lines, in one direction shall be called positive distances, and shall be denoted by the sign+; distances measured in the opposite direction shall be called negative distances, and shall be denoted by the sign.

The convenience of this custom, fashion, or rule, will become apparent in the examples that follow.* In Fig. 31 let distances

* Advances in mathematics have often depended upon the introduction of a good custom which has at last been universally adopted and made a rule. Thus, for example, the custom of using exponents to show the power to which a quantity is raised, which was first introduced in the first half of the sixteenth century, and made gradual progress until its final establishment in the latter half of the seventeenth century, has been of great service in aiding the advances of algebra.

measured in the direction of N be taken positively; then distances measured in the direction of M will be taken negatively. On directed lines the direction in which a line is measured, or in which a point moves on a line, is indicated by the order of the letters naming the line. Thus, for example, if a point moves from B to C, the distance passed over is read BC. In this reading, the starting point is indicated by the first letter B, and the stopping point, by the last letter C. After the same fashion, CB means the distance from C to B. If, for instance, there are 3 units of length between B and C, then BC=+ 3, CB =

-=

3.

EXAMPLES.

1. Suppose a point (Fig. 31) moves from P to B, thence to C, thence to D, thence to F. Let the number of units of length between P and B, B and C, C and D, F and D, be 2, 3, 2, 10, respectively. The point starts at P and stops at F; hence the distance from the starting point to the stopping point is PF. In this case the point's trip from P to F is made in several steps as indicated above. That is, on properly indicating the lines passed over,

This shows that the final position of the moving point is three units to the left of P. This example also shows one great convenience of the rule of signs in measurement, namely, that by attending to this rule and to the proper naming of the lines passed over by a moving point, one immediately obtains the result of the successive movements.

NOTE. In the following examples, in lines that lie east and west, let measurements toward the east be taken positively; in lines that lie north and south, let measurements toward the north be taken positively.

2. A man travelling on an east and west line goes east 20 mi., then east 16 mi., then west 18 mi., then east 30 mi. What is his final distance from the starting point? [Draw a figure, and indicate the successive trips by letters.]

3. A man travelling on an east and west line goes west 20 mi., then east 10 mi., then east 25 mi., then east 30 mi., then west 45 mi. Do as in Ex. 2.

4. A man travelling on a north and south line goes north 100 mi., then south 60 mi., then south 110 mi., then north 200 mi., then north 15 mi., then south 247 mi. Do as in Ex. 2.

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.

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 re-
volving from the position OX and stops
its revolution at the position OQ. The
line OX at which the revolution begins, X,
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.

Terminal Line

Initial Line X

FIG. 32.

Let YOY, be at right angles to XOX. 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 OX1, 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

P

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

Q

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°, 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, X1OY1 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.