the trigonometric functions, but before we proceed to do so we shall extend our ideas of angles beyond the knowledge we obtained of them in the elementary geometry. There an angle is defined by some such definition as the following: the plane figure formed by two straight lines drawn from the same point. The unit of angles is either the right angle, or the degree, and the largest angle usually dealt with is equivalent to two right angles and is often called a straight angle. In trigonometry, on the other hand, we deal with angles of any magnitude whatever. To do so we introduce the idea of motion, of revolution. Starting from the initial position OX, Fig. 1, we may revolve the line about O in the direction indicated by the arrows, stopping in any desired terminal position OP1, OP2, OP3, OP, etc. In this way angles of any number of degrees whatever may be gen erated. Thus, if we stop in the position OY, we have an angle of 90°; in the position OX', 180°; in the position OP, 225°; in the position OY', 270°, and so on. By making one whole revolution we should arrive at an angle of 360°; two and one half revolutions, 900°; etc. Not only so, but we might revolve from the initial position OX in the opposite direction. Now oppositeness is indicated algebraically by the use of the signs plus (+) and minus (−). So that if we agree to take the positive direction of revolution counterclockwise, then clockwise will be the negative direction and we can thus generate negative angles of any magnitude whatever. Thus, Fig. 1, the angle XOP, is 225° if we have revolved in the positive direction, but is 135° if we have revolved in the negative direction. When an angle lies in value between 0° and 90° it is said to be an angle in the first quadrant since its terminal side lies in the first quadrant. An angle lying in value between 90° and 180° is said to be in the second quadrant; between 180° and 270°, in the third quadrant; between 270° and 360°, in the fourth quadrant. If A is a positive angle in the first, second, third, or fourth quadrant respectively, add graphically 3. Abscissa, Ordinate, and Distance. Consider an angle, positive or negative, of any magnitude whatever*, XOP, of Fig. 2. FIG. 2. From P, any point in the terminal side of this angle, drop a perpendicular upon the axis of x. The lines of the figure are named as follows: OM is called the abscissa of the point P, MP the ordinate, and OP the distance. The abscissa OM and the ordinate MP are together called the coördinates of the point P. Note very carefully that the abscissa is always read from 0 to M, the ordinate from M to P; that is, in each case from the axis to the * As a matter of convenience we do not consider angles numerically greater than 360°. It is obvious that the discussion applies equally well point. The distance is read from 0 to P. Thus, for an angle in the second or third quadrant the direction of the abscissa is opposite to that of an angle in the first or fourth quadrant. For an angle in the third or fourth quadrant the direction of the ordinate is opposite to that of the ordinate of an angle in the first or second quadrant. Oppositeness in direction being distinguished as usual by difference in algebraic sign we have the following conventions: The abscissa measured to the right of the axis of y is positive; to the left, negative. The ordinate measured upward from the axis of x is positive; downward, negative. The distance is measured from the origin outward and is taken positive. EXAMPLES 1. The abscissa of a point is 3, its ordinate 4; find the distance. 2. The distance of a point is 5, its ordinate 4; find the abscissa 3. The ordinate of a point is 2, its distance 3; find the abscissa. 4. The ordinate of a point is -5, its abscissa -4; find the distance. 5. Prove that the square of the distance of any point is equal to the sum of the squares of the abscissa and ordinate. 6. Prove that for all points on a straight line through the origin the ratio of the ordinate to the abscissa is constant. 4. The Trigonometric Functions Defined. Let us now proceed to define the six trigonometric functions of an angle; six quantities which depend upon the angle for their values. They are the possible ratios between the various pairs of the three lines named in Art. 3. Thus, Fig. 2, the tangent XOP ordinate of P_MP Three other functions are sometimes used: The versed sine, which is unity minus the cosine; the coversed sine, which is unity minus the sine; the suversed sine, which is unity plus the cosine. They are relatively unimportant. 5. Trigonometric Functions are Ratios. The first thing to be noted about these functions is that, being ratios, they are independent of the actual lengths of the abscissa, ordinate, and distance. Thus, Fig. 3, the triangles OM,P1, OM2P, and OMP being similar, their homologous sides are proportional, so that Similarly the truth of the statement may be shown for 6. Signs of the Functions. The second point to be noted is that the signs of the functions vary according to the quadrant in which the angle lies. Thus, Fig. 2, for the angle XOP in the first quadrant the abscissa, ordinate and distance are all positive so that all the functions are positive. For the angle XOP in the second quadrant the ordinate and distance are positive, the abscissa negative. Thus we have for the angle in the second quadrant The following table gives the signs of the functions in 7. Let the student determine, as above, the signs of the trigono |