133. We proceed to show that the Trigonometrical Ratios of an angle vary in Sign according to the Quadrant in which the angle happens to be.

From the definition we have, with the usual letters,

I. When ROP lies in the first Quadrant (Fig. I.). MP is positive because from M to P is upwards

(Rule II. p. 95.)

OM is positive because from 0 to Mis towards the right

OP is positive

(Rule 1.)

(Rule III.)

Hence, if A be any angle in the first Quadrant,

II. When ROP lies in the second Quadrant, (Fig. 11.).
MP is positive, because from M to P is upwards,

OM is negative, because from 0 to M is towards the left.
OP is positive.

Hence, if A be any angle in the second Quadrant,

III. When ROP lies in the third Quadrant (Fig. III.)

MP is negative, OM is negative, OP is positive.

So that, if A be any angle in the third Quadrant,
sin A is negative, cos A is negative, tan A is positive.

IV. When ROP lies in the fourth Quadrant (Fig. IV) MP is negative, OM is positive, OP is positive.

So that, if A be any angle in the fourth Quadrant,
sin A is negative, cos A is positive, tan A is negative.

The student should notice that in any particular Quadrant the three signs of sine, cosine, and tangent are unlike their signs in any other Quadrant.

EXAMPLES. XXV.

State the sign of the sine, cosine, and tangent of each of the following angles :

135. The NUMERICAL VALUES through which the Trigonometrical Ratios pass, as the angle ROP moves through the first Quadrant, are repeated as the angle ROP moves through each of the other Quadrants.

Thus as P moves through the second Quadrant from U to L, Fig. II, (OP being always of the same length) MP and OM pass through the same succession of numerical values through

which they pass, as P moves through the first Quadrant in the opposite direction from U to R.

Example 1. Find the sine, cosine and tangent of 120o.

120o is an angle in the second Quadrant.

Let the angle ROP be 120o (Fig. II. p. 96.)

Then the angle POL=180o – 120o= 60o.

MP

Hence, sin 120°— =sin 60° numerically, and in the second

OP

OM

OP

(i).

Again, cos 120o = =cos 60o, numerically, and in the second

Quadrant the cosine is negative.

Example 2. Find the sine, cosine and tangent of 225o.

225o is an angle in the third Quadrant.

Let the angle ROP be 225° (Fig. III.)

(ii).

(iii).

Here the angle POL=225o - 180°=45o.

Therefore the Trigonometrical Ratios of 225o those of 45o numerically; and in the third Quadrant the sine and cosine are each negative and the tangent is positive.

136. The cosecant, secant and cotangent of an angle A have the same sign as the sine, cosine, and tangent of A respectively.

EXAMPLES. XXVI.

Find the algebraical value of the sine, cosine and tangent of the following angles :

*Example. To trace the changes in the magnitude and sign of sin A, as A increases from 0° to 360°.

Take the figure and construction of page 96.

As A increases from 0° to 90°, MP increases from zero to OP, and is positive.

Therefore sin A increases from 0 to 1 and is positive.

As A increases from 90° to 180°, MP decreases from OP to zero, and is positive.

Therefore sin A decreases from 1 to 0 and is positive.

As A increases from 180° to 270°, MP increases from zero to OP, and is negative.

Therefore sin A increases numerically from 0 to 1 and is negative.

As A increases from 270° to 360o, MP decreases from OP to zero, and is negative.

Therefore sin A decreases numerically from 1 to 0 and is negative.

* EXAMPLES. XXVII.

Trace the changes in sign and magnitude as A increases from 0° to 360° of