CHAPTER II. USE OF THE SIGNS + AND -. DEFINITIONS OF THE TRIGONOMETRICAL FUNCTIONS AND RELATIONS CONNECTING THEM. TRIGONOMETRICAL FUNCTIONS OF CERTAIN FIXED ANGLES. 1. Use of the signs + and to indicate contrariety of direction. Algebraical representation of straight lines and angles. Let O be a fixed point in a straight line X'OX, and let OA contain a units of length, and let AB contains b units, X' + B A X AB being first supposed less than OA. Then OB will contain a-b units of length. If we suppose any assigned point A to be reached by a point starting from O and travelling along the line OX, B will be determined by the point travelling back from A over b units of length. Suppose now that AB is greater X' than OA, B will in this case The necessity however of making any distinction between the cases in which b is less or greater than a, can be obviated if we agree to represent a distance measured to the left of O by a symbol with a negative sign prefixed. For, in accordance with this convention, b-a being the magnitude of the distance OB, the position of the point B will be indicated by —(b-a); i. e. a−b, as in the case in which b is less than a. It is this generality of algebraical representation, this power of including all the possible cases of a theorem under one algebraical formula, which constitutes the principal utility of algebraical analysis as applied to geometry. It will be readily seen that this same method of symbolical representation may be applied to angles, with reference to the direction in which a revolving line must move from its initial position in order that it may come into coincidence with any assigned position. Thus, if OX be taken for the initial line, if the angle XOK be represented algebraically by +a, the angle XOK', which is drawn equal in magnitude to XOK, will be represented by -α. K α X -α K' For the revolving line, in order to pass over the angle XOK', must revolve in a direction opposite to that in which it moves in passing over the angle XOK. 2. Position of a point in space and of the revolving line. Let X'OX, YOY' be two straight lines at right angles to each other. The position of any point in space can be determined with reference to O by means of the above methods of representing the position and magnitude of straight lines. Suppose we agree to affix the positive sign to the symbols which represent the lengths of lines measured from 0 in the directions OX, OY; then if we know the proper symbols representing the distances of any point from X'OX and Y'OY respectively, the position of the point is determined. Thus + a +b indicate a point P, in the quarter XOY, N.B. It is customary to represent lines measured on XX' by symbols, to which the positive sign is prefixed, when they are measured from O towards the right, and consequently lines measured to the left of O by symbols affected with the negative sign. Also the direction OY above XX' is that which is generally taken to correspond to symbols which have the positive sign in representing lines measured on YY'. Similarly, symbols representing angles when affected with the sign + are generally supposed to represent angles traced by a revolving line, which moves from coincidence with the initial line in a direction contrary to that of the hands of a watch. Such arrangements are of course perfectly arbitrary. It is manifestly a slovenly and inaccurate form of expression to talk of positive or negative lines, positive or negative directions of revolution. A line or a direction cannot be positive or negative, inasmuch as they are ideas which are strictly geometrical, and plainly quite distinct from the conceptions of algebraical representation with which they are associated. To obviate this objection, therefore, I propose to call the directions OX, OY the primary, and OX', OY' the secondary directions of measurement. Similarly, I shall call the direction of revolution opposite to that of the hands of a watch the primary, and the opposite one the secondary direction of revolution. The angle XOK I shall call a primary angle, and the angle XOK' a secondary angle. 3. Definitions of the Trigonometrical Ratios. Let OX be the initial line from coincidence with which a line OP revolves in sweeping over any angle. Let OP be any position of the revolving line which will correspond to an angle whose symbol will be positive or negative according to the direction in which OP is supposed to move from coincidence with OX. Let any point P be taken in OP and from it let a perpendicular be dropped upon the initial line (produced backwards, as OX', if necessary). Take proper algebraical symbols to represent the sides of the right-angled triangle PON so formed, and call the angle through which OP has revolved A. The ratio of the algebraical representative of PN to that of OP* is called the sine of A, ..ON ..PN ОР cosine of A, tangent of A. The reciprocals of the sine, cosine and tangent are called the cosecant, secant and cotangent of A respectively. It is evident that the sine and cosine can neither of them exceed unity in magnitude, since the perpendicular and base are neither of them ever greater than the hypotenuse of a right-angled triangle. Hence the cosecant and secant can never be less than unity. The defect of the cosine from unity is called the versed sine. The versed sine of A is generally written versin A. Twice the sine of the half of A is called the chord of A, for a reason which will be hereafter explained, and is written chd A. The definitions of the trigonometrical ratios (or functions as they are also called) which are ordinarily given are something as follows: "If from a point in one of the straight lines containing an angle a perpendicular be let fall upon the other side, or the other side produced, the ratio of the perpendicular to the hypotenuse is called the sine of the angle." * The symbol representing a line measured from O along OP is taken with a positive sign. If the sign were negative, then, in accordance with the rules of algebraical representation, the line indicated would be one measured from O along OP produced backwards, i. e. in the direction PO. |