20 We will proceed to discuss these more fully. An angle is formed by a line starting from a fixed position and revolving in the direction opposite to the hands of a watch. The point P moves in a circle and PAX increases to PAX. From P a perpendicular PM is let fall on the fixed line. As the angle increases, PM increases, and AM diminishes; while P moves 21 the angle. So too if we know the ratio of AM to AP ; for instance, if AP=2AM the size of the angle PAM is fixed. Just as in circular measure we examine the fraction which the arc is of the radius, how many radii or parts of radii there are in the arc, so here we enquire what fractions the perpendicular PM and the base AM are of the radius. The first of these is called the sine of A. 22 23 The other is called the cosine of A. cos A = AM and decreases when the angle increases. There are six other ratios of the angle PAM, which are derived from these, and they are of two kinds; those which increase while the angle increases and decrease while it decreases, and those which vary in the opposite way, viz. decrease while the angle in creases. The first of these are the sine, the tangent, the secant, and the versed sine of A. The others are of the opposite kind, they decrease when the angle increases. These are distinguished by the prefix co, since they are the corresponding ratios of the complement of A; they are the cosine, the cotangent, the cosecant, and the coversed sine of A. Now these are all connected with sine and cosine of A in a way that should thus be remembered: versin A=1-cos A; covers A=1- sin A. It is best to remember the principle of increase and decrease, and so to have the ratios sorted into the 2-2 24 two corresponding sets; the one direct and the other inverse. The tangent and cotangent of A are the ratios of the sides which contain the right angle. So the numerator must be in one case the increasing and in the other the decreasing side. If a line AM of four parts have a perpendicular PM erected on it of three such parts, an angle will be formed by AP with AM, whose Take a line AP divided into five equal parts, and take a line PM equal to three of these parts. A circle described on diameter AP in which the line PM is placed will On the use of these three formulæ and the original connections of tan A and cot A with sin A and cos A, and of sec A and cosec A with sin A and cos A, some exercises will be given. In proving these identities, one side should be taken, and reduced to the form of the other by the aid of these three formulæ and the original connections. (1) Example. Prove tan2 A = sec2 A — 1. |