CHAPTER II.

THE TRIGONOMETRIC FUNCTIONS.

§ 22. A right triangle can always be formed by letting fall a perpendicular from any point in the terminal line of a given angle upon the initial line, produced if necessary. This triangle we will call a triangle of reference for the angle; and, in applying the rules of § 20 to determine the signs of its sides, we shall always regard the hypothenuse and base as measured from the vertex of the given angle, and the perpendicular as measured from the initial

line.

In each of the figures 22, 23, 24, and 25, for example,

where (XOP) is, in each case, the given angle, we can form the right triangle оCB, as stated above. In order to form this triangle in figures 23 and 24, we must produce the line ox in the direction ox'.

In the above figures, the letters x, y, and r represent the base (oc), the perpendicular (CB), and the hypothenuse (OB) respectively, both in direction and length. Therefore x is positive in figures 22 and 25, and negative in figures 23 and 24; y is positive in figures 22 and 23, and negative in figures 24 and 25; 7 is positive in each figure. But might be made negative in each figure. Thus, in Fig. 22, we may take x=oc", y=c′′B′′, r= OB", making x, y, and r all negative.

§ 23. The Trigonometric Functions of an angle are the six ratios between the three sides of a triangle formed as above, taking into account the signs as well as the lengths of these sides.

These ratios are always the same for the same angle,

both in numerical value and in sign; for if, as in figure 22, we let fall perpendiculars from different points of the terminal line, as B, B', and B", we have the right triangles OCB, OC'B', and oc"B", which evidently have their acute angles at o equal. These triangles are therefore (v. § 21, d) similar, and their homologous sides are proportional; i.e., the six ratios of the sides of OCB are, in numerical value, respectively equal to the six ratios of the sides of ос'B', or Oс"B". They are equal in sign as well; for, in comparing any two of the triangles, the sides of the first either have the same directions, and therefore the same signs, as the corresponding sides of the second, as in OCB and oc'B'; or, as in OCB and oс"B", each side of the second has a direction, and therefore a sign, opposite to that of the corresponding side of the first; and in each case the signs of the ratios are obviously the

same.

These ratios are, in general, different for different angles; for two right triangles, not having an acute angle of the one equal to an acute angle of the other, are not similar, and their homologous sides are not proportional.

Thus we see that the values of the trigonometric ratios of an angle depend, and depend only, upon the value of the angle; for this reason they are called the trigonometric functions of the angle.

*

As soon as we have given names to these ratios, we shall be able to introduce into the same formula two entirely different kinds of quantities, -angles and straight lines. We shall then have made a very important step in the treatment of our subject.

* When one quantity depends upon another for its value, so that a change in the second necessitates a change in the first, the first is called a function of the second.

§ 24. To each of the six ratios mentioned above, a name has been given as follows: In each of the figures

§ 25. Bearing in mind what has been said about the signs of x, y, and r, in § 22, we easily deduce the following table for the signs of the functions of angles in each quadrant:

Since the hypothenuse of a right triangle is the greatest of the three sides, the first two of our six ratios never exceed 1 in numerical value, and the last two never fall below 1 in numerical value. Hence the sine and cosine vary only from 1 to +1; the secant and cosecant vary only from +1 to +∞o and from — 1 to the tangent and cotangent may have any value from ∞

∞; while

Therefore the sine, cosine, and tangent are reciprocals

of the cosecant, secant, and cotangent respectively.

Any two of the quantities appearing in either [2] or

[3] being given, it is evident that the third can be found.