That is, if the angles AOB and AOB' are each equal to 30° in absolute value, we should say that AOB = +30°, and - AOB': - 30°.

We may thus conceive of negative angles of any magnitude whatever. It is immaterial which direction we consider the positive direction of rotation; but having adopted a certain direction as positive, our subsequent operations must be in accordance.

24. The fixed line 04, from which the rotation is sup-, posed to commence, is called the initial line, and the final position of the rotating radius is called the terminal line.

Either side of an angle may be taken as the initial line, the other being then the terminal line; thus, in the angle AOB, we may consider OA the initial line and OB the terminal line, in which case the angle is positive; or we may consider OB the initial line and OA the terminal line, in which case the angle is negative.

25. In designating an angle, we shall always write first. the letter at the extremity of the initial line; thus, in designating the angle AOB, if we regard OA as the initial line, we should call it AOB, and if we regard OB as the initial line, we should call it BOA.

26. There are always two angles in absolute value less than 360°, one positive and the other negative, formed by a given initial and terminal line.

Thus, there are formed by OA and OB' the positive angle AOB' greater than 270°, and the negative angle AOB' less than 90°.

We shall distinguish between such angles by referring to them as "the positive angle AOB'," and "the negative angle AOB'," respectively.

27. It is evident that the terminal lines of two angles which differ by a multiple of 360° are coincident; thus, the angles 30°, 390°, - 330°, etc., have the same terminal line.

28. Let XX' and YY' be a pair of straight lines at right angles to each other; let P1 be any point in their plane, and draw PM perpendicular to XX'.

The distances OM and PM are called the rectangular coordinates of P1 with reference to the lines XX' and YY'.

OM is called the abscissa, and PM the ordinate; the lines XX' and YY' are called the axes of X and Y, respectively, and their intersection O is called the origin.

=

29. If in the above figure, OM ON=b, and the perpendiculars P1M=P2N=P ̧N= P1M= a, the points P1, P2, P3, and P will have the same co-ordinates.

To avoid this ambiguity, the following conventions have been adopted:

Abscissas measured to the right of O are considered positive, and to the left, negative.

Ordinates measured above the line XX' are considered positive, and below, negative.

Then the co-ordinates of the four points will be:

Note. In the figures of this chapter, the small letters denote the lengths of the lines, without regard to their algebraic sign.

30. It is customary to denote the abscissa and ordinate of a point by the letters x and y, respectively. Thus the fact that the abscissa of a point is equal to b and its ordinate to a, is expressed by the saying that for the point in question x=b and y:

= a.

The same fact may be stated more concisely by referring to the point as "the point (b, a)", where the first quantity in the parenthesis is understood to be the abscissa, and the second the ordinate.

31. If a point lies upon the axis of X, its ordinate is zero; and the same is true of the abscissa of a point upon the axis of Y.

GENERAL DEFINITIONS OF THE FUNCTIONS.

32. We will now give general definitions for the trigonometric functions, applicable to any angle whatever.

Take the initial line as the positive direction of the axis of X, the vertex being the origin.

Take any point in the terminal line, and construct its rectangular co-ordinates by dropping a perpendicular to the initial line, produced if necessary.

Then, designating the distance of the assumed point from the origin as the "distance" of the point,

The SINE
The COSINE
The TANGENT

is the ratio of the ORDINATE to the DISTANCE. is the ratio of the ABSCISSA to the DISTANCE. is the ratio of the ORDINATE to the ABSCISSA. The COTANGENT is the ratio of the ABSCISSA to the ORDINATE. is the ratio of the DISTANCE to the ABSCISSA.

The SECANT

The COSECANT

is the ratio of the DISTANCE to the ORDINATE.

Note. These definitions include those of Art. 10. The definitions of the versed sine and coversed sine, given in Art. 11, are sufficiently general to apply to any angle whatever.

33. We will now apply the definitions of Art. 32 to the angles XOP1, XOP2, XOP, and XOP, in the following figures:

4

Let P1, P2, P3, and P1 be any points on the terminal lines OP1, OP, OP, and OP, and let their co-ordinates be (b, a), (—b, a), (—b, − a), and (b, - a), respectively.

34. Since the terminal lines of two angles which differ by a multiple of 360° are coincident (Art. 27), it is evident that the trigonometric functions of two such angles are identical.

Thus, the functions of 50°, 410°, 770°, - 310°, etc., are identical.

• It is customary to express this fact by saying that the trigonometric functions are periodic.