Then it is universally agreed to consider that

(1) horizontal lines to the right of YY' are positive, horizontal lines to the left of YY' are negative;

(2) vertical lines above XX' are positive,

vertical lines below XX' are negative.

Thus OM, OM, are positive, OM, OM, are negative;

3

M1P1, MP, are positive, MP3, MP are negative.

2

74. Convention of Signs for Angles.

In Art. 2 an angle

has been defined as the amount of revolution which the radius vector makes in passing from its initial to its final position.

In the adjoining figure the straight line OP may be supposed to have arrived at its present position from the position occupied by OA by revolution about the point 0 in either of the two directions indicated by the arrows. The angle AOP may thus be regarded in two senses according as we suppose the revolution to have been in the same direction as the hands of a clock or in the opposite direction. To distinguish between these cases we adopt the following convention:

A

when the revolution of the radius vector is counter-clockwise the angle is positive,

when the revolution is clockwise the angle is negative.

H. K. E. T.

5

Trigonometrical Ratios of any Angle.

75. Let XX' and YY' be two straight lines intersecting at right angles in O, and let a radius vector starting from OX revolve in either direction till it has traced out an angle A, taking up the position OP.

From P draw PM perpendicular to XX'; then in the rightangled triangle OPM, due regard being paid to the signs of the lines,

The radius vector OP which only fixes the boundary of the angle

is considered to be always positive.

From these definitions it will be seen that any trigonometrical function will be positive or negative according as the fraction which expresses its value has the numerator and denominator of the same sign or of opposite sign.

76. The four diagrams of the last article may be conveniently included in one.

With centre 0 and fixed radius let a circle be described; then the diameters XX' and YY' divide the circle into four quadrants XOY, YOX', X'OY', Y'OX, named first, second, third, fourth respectively.

Let the positions of the radius vector in the four quadrants be denoted by OP1, OP2, OP3, OP4, and let perpendiculars P1M1, P2M2, P3M3, P4M4 be drawn to XX'; then it will be seen that in the first quadrant all the lines are positive and therefore all the functions of A are positive.

2

In the second quadrant, OP, and MP2 are positive, OM2 is negative; hence sin A is positive, cos A and tan A are negative.

In the third quadrant, OP, is positive, OM, and MP3 are negative; hence tan A is positive, sin A and cos are negative.

In the fourth quadrant, OP, and OM, are positive, MP4 is negative; hence cos A is positive, sin A and tan A are negative.

77. The following diagrams shew the signs of the trigonometrical functions in the four quadrants. It will be sufficient to consider the three principal functions only.

The diagram below exhibits the same results in another useful form.

78. When an angle is increased or diminished by any multiple of four right angles, the radius vector is brought back again into the same position after one or more revolutions. There are thus an infinite number of angles which have the same boundary line. Such angles are called coterminal angles.

If n is any integer, all the angles coterminal with A may be represented by n. 360°+4. Similarly, in radian measure all the angles coterminal with may be represented by 2nя+0.

From the definitions of Art. 75, we see that the position of the boundary line is alone sufficient to determine the trigonometrical ratios of the angle; hence all coterminal angles have the same trigonometrical ratios.

and

1

For instance, sin (n. 360°+45°)=sin 45°: ;
√2

Example. Draw the boundary lines of the angles 780°, -130°, - 400°, and in each case state which of the trigonometrical functions are negative.

(1) Since 780=(2 × 360) +60, the radius vector has to make two complete revolutions and then turn through 60°. Thus the boundary line is in the first quadrant, so that all the functions are positive.

(2) Here the radius vector has to revolve through 130° in the negative direction. The boundary line is thus in the third quadrant, and since OM and MP are negative, the sine, cosine, cosecant, and secant are negative.

(3) Since -400 - (360+40), the radius vector has to make one complete revolution in the negative direction and then turn through 40°. The boundary line is thus in the fourth quadrant, and since MP is negative, the sine, tangent, cosecant, and cotangent are negative.

M

P

M

EXAMPLES. VIII. a.

State the quadrant in which the radius vector lies after describing the following angles:

For each of the following angles state which of the three principal trigonometrical functions are positive.

15.

16.

17.