Example. (180° - A) indicates

(i) the angle described by OP turning about O from the position OR in the positive direction until it has described an angle of (180 - A) 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 180o (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 O 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.

(12) (2n+1) π −7. (13) 2nπ-. (14) (2n+1) π −7

2

Note. n always stands for a whole number of two right angles.

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.
UL is called the second Quadrant.

LD is called the third Quadrant.
DR is called the fourth 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. 250; 385° i.e. 360° +25°; 745o i.e. 2 × 360o + 25o ; - 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 90o, and let n be any whole number, positive or negative.

Then

(i) 2n x 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 × 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) × 180o, OP is in the position OL.] The corresponding expressions in circular measure are

(i) 2n +0; (ii) 2n π-0; (iii) (2n+1) π-0;
(iv) (2n+1) π + 0.

EXAMPLES. XXIV.

State in which Quadrant the revolving line will be after

131. The principal directions of lines with which we are concerned in Trigonometry are as follows;

i. that parallel to the initial line OR (OR is usually drawn from 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.

Accordingly we make the following rules:

I. Any line drawn parallel to LR in the direction from left to right is to be positive; and consequently (Art. 112) any line drawn parallel to RL in the opposite direc tion, i.e. from right to left, is to be negative.

II. Any line drawn parallel to DU in the direction from D to U, upwards, is to be positive; and consequently any line drawn parallel to UD in the opposite direction, i.e. downwards, is to be negative.

III. Any line drawn parallel to the revolving line in the direction from 0 to P is to be positive, and consequently any line drawn in the direction from P to O is to be negative.

Note. The student must notice that the revolving line OP carries its positive direction round with it, so that the line 'OP' is always positive.

132. We said, in Art. 81, that the definitions of the Trigonometrical Ratios (on pp. 46, 47), apply to angles of any magnitude. We have only to remark that it is generally convenient to take P on the revolving line; that PM is drawn perpendicular to the other line produced if necessary; and that the order of the letters in MP, OP, OM is an essential part of the definition.

The order of the letters P, M, O in the expressions

MP

OP'

etc., is therefore of great importance.