CHAPTER IX. ON THE USE OF THE SIGNS + AND IN TRIGONOMETRY. 126. IN Trigonometry in order conveniently to treat of angles of any magnitude, we proceed as follows. We take a fixed point O, called the origin; and a fixed straight line OR, called the initial line. The angle of which we wish to treat is described by a line OP, called the revolving line. This line OP starts from the initial line OR, and turns about through an angle ROP of any proposed magnitude into the position OP. 127. We have already said in Art. 41 (i) that, when an angle ROP is described by OP turning about O in the direction contrary to that of the hands of a watch, the angle ROP is said to be positive; that is, is represented algebraically by its measure with the sign + before it. (ii) that, when an angle ROP is described by OP turning about O in the same direction as the hands of a watch, the angle is said to be negative; that is, is represented algebraically by its measure with the sign - before it. Example. (1800 - A) indicates (i) the angle described by OP turning about 0 from the position OR in the positive direction until it has described an angle of (1804) degrees. Or, (ii) the angle described by OP turning about 0, from the position OR, in the positive direction until it has described an angle of 180° (when it has turned into the position OL), and then turning back from OL in the negative direction through, the angle - A into the position OP. Or, (iii) the angle described by OP turning about 0 from the position OR, in the negative direction through the angle - A, and then turning back in the positive direction through the angle 180°, into the position OP. The student should observe that in each of these three ways of regarding the angle (180°-A), the resulting angle ROP is the same. EXAMPLES. XXIII. Draw a figure giving the position of the revolving line after it has turned through each of the following angles. π (12) (2n+1)- -. (13) 2nπ- (14) (2n+1) π − π 2 π 2 128. It is often convenient to keep the revolving line of the same length. In this case the point P lies always on the circumference of a circle whose centre is 0. Let this circle cut the lines LOR, UOD in the points L, R, U, D respectively. The circle RULD is thus divided at the points R, U, L, D into four Quadrants, of which RU is called the first Quadrant. LD is called the third Quadrant. Hence, in the figure, ROP, is an angle in the first Quadrant. 129. When we are told that an angle is in some partiticular Quadrant, say the third, we know that the position in which the revolving line stops is in the third Quadrant. But there is an unlimited number of angles having this same final position of OP. Example. 25°; 385° i.e. 360° +25°; 745° i.e. 2 × 360° +25°; - 335 i.e. - 360° +25° are each an angle in the first Quadrant, and are all represented geometrically by the same final position of OP. 130. Let A be an angle between 0° and 90°, and let n whole number, positive or negative. be Then any (i) 2n × 180° + A represents algebraically an angle in the first Quadrant. (ii) 2n x 180°-A represents algebraically an angle in the fourth Quadrant. [For 2n x 180° represents some number n of complete revolutions of OP; so that after describing n × 360o, OP is again in the position OR.] (iii) (2n + 1) × 180° – A represents algebraically an angle in the second Quadrant. (iv) (2n + 1) × 180° + A represents algebraically an angle in the third Quadrant. [For after describing (2n+1) x 180o, OP is in the position OL.] (i) 2n +0; (ii) 2nπ-0; (iii) (2n+1) π-0; EXAMPLES. XXIV. State in which Quadrant the revolving line will be after describing the following angles : 131. The principal directions of lines with which we are concerned in Trigonometry are i. that parallel to the initial line OR (OR is usually drawn from 0 towards the right hand, parallel to the printed lines in the page; and RO is produced to L.) ii. that parallel to the line DOU, which is drawn through O at right angles to LOR; iii. that parallel to the revolving line OP. |