Right motion; so that the negative direction is the same as that of the motion of the hands of a clock; i.e. a Left-Up-Right-DownLeft motion.

(2) The initial line shall be supposed to be drawn rightwards.

Since a line drawn upwards makes (in accordance with this convention) a positive right-angle with a line drawn right-wards, we have, combining the Mathematician's with the Teacher-andStudent's convention, the following three positive directions: (1) For initial line, Right-wards. (2) For revolution, Right-upwards. (3) For perpendicular, Upwards.

304. To determine the signs of the ratios for the different quadrants in which the final line may lie.

Let OI be the initial line, OF the final line.

For simplicity, take the hypothenuse (OH) in every case along OF. So that (OH) is always positive.

Draw HB perpendicular to 01 or OI produced.

Then (OB) is the base; and (BH) is the perpendicular.

Let (OK) make a positive right-angle with (OI), so that lines in the direction (OK) are positive.

(1) When OF is in the 1st quadrant,

(BH) which is drawn in the direction (OK) is Positive.

(BH)

(OB)

oppositely to (OK) is Negative.
oppositely to (01) is Negative.

(4) When OF is in the 4th quadrant,

Thus the signs corresponding to the four quadrants are

for Perp.; (1)+; (2) + ; (3) − ; (4)—.

for Base; (1) +; (2) − ; (3) − ; (4) +.

Now since we have taken the hypothenuse (OH) to be positive in every case,

(I) Sine and Cosecant have the sign of the Perp.

(II) Secant and Cosine have the sign of the Base. Also (III) Tangent and Cotangent have the sign obtained by division of Base and Perp.

The following table gives all the results:

Here observe that, while every ratio is positive in the 1st quadrant,

the sine (and cosecant) are also positive in the 2nd;

(1) The initial line OI is drawn Rightwards:

(2)

so that

that

The positive direction of revolution is Right-upwards:

(3) OK, which is drawn Upwards, is positive.

Also the lines (OB), (BH) are dotted when negative.

Retaining these figures in the mind, it is easy to remember

(OB) is +ve, when Rightwards; - ve, when Leftwards.
(BH) is +ve, when Upwards; ve, when Downwards.

306. But the student must particularly observe, that the signs here ascribed to the base and perpendicular, according as the former is rightwards or leftwards and the latter upwards or downwards, apply only to the angle IOF in each of the four figures.

Thus, in each figure, when we regard (OB) as the base of the acute angle BOH, it is positive: so that (e.g.) cos (BOH) is positive in each case.

Moreover, retaining the same positive direction of revolution viz. Right-Up-Left-Downwards,

in 2nd figure, sin (BOH) is negative.

in 3rd figure, sin (BOH) is positive.

Comparison of the ratios of related angles.

307. In denoting the ratios of angles, we may use either symbols for directed lengths, or symbols for mere Arithmetic lengths -prefixing explicitly the proper signs.

The latter method was used in Art. 136, in defining the ratios of obtuse angles. The student ought to be able to apply either method when required.

308. In each of the figures of Art. 304, let BOH represent the same positive acute angle A.

Then the ratios of

BOH are all positive.

Now, in the first figure,

IOF represents the positive angle A,

or generally the angle 2n. 180° + A.

In the second figure,

IOF represents the positive angle

180° – A, or generally the angle (2n + 1) 180° – A.

In the third figure, IOF represents the positive angle 180° + A, or generally the angle (2n + 1) 180° + A.

In the fourth figure, IOF represents the negative angle - A, or generally the angle 2n. 180° - A.

Thus any ratio has the same numerical value for all the angles found by adding or subtracting A from 0 or 180° or any multiple of 180°.

But the sign of the ratio has to be found by considering in what quadrant the final line would lie for the angle in question.

309. The above results may be expressed as follows:

If A is an acute angle, and n any integer.

=

that same ratio of A.

I. Any ratio of 2n. 180° + A II. The sine or cosecant of (2n + 1) 180° – A is the same as that of A; but its other ratios are the negatives of those of A.

III. The tangent or cotangent of (2n+1) 180° + A is the same as that of A; but its other ratios are the negatives of those of A.

IV. The secant or cosine of 2n. 180° - A is the same as that of A; but its other ratios are the negatives of those of A.

310. The above results hold whatever value A may have. This may be shown in the following way.

The revolution 2n. 180° carries the revolver from OI back again to OI.