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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.

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

Let (OK) make a positive right-angle with (01), 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.
(OB)

in the direction (01) is Positive.
(2) When OF is in the 2nd quadrant,
(BH)

in the direction (OK) is Positive.
(OB)

oppositely to (OI) is Negative.
(3) When OF' is in the 3rd quadrant,
(BH)

oppositely to (OK) is Negative. (OB)

oppositely to (OI) is Negative. (4) When OF is in the 4th quadrant, (BH)

oppositely to (OK) is Negative. (OB)

in the direction (01) is Positive. 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:

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

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

in the 3rd ;
the secant (and cosine)

in the 4th. 305. In the above figures, (1) The initial line 01 is drawn Rightwards : () The positive direction of revolution is Right-upwards :

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so that

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(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 that

(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, 2 IOF represents the negative angle – A, or generally the angle 2n. 180° – A. Thus, from first figure,

any ratio of 2n. 180° + A that same ratio of A.
From second figure,

BH
sin (180° - A) = sin A.

OH

ОВ
cos (180° – A)=

COS A.
OH

BH
tan (180o – A)=

- tan A. – ОВ

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COS A.

= COS A.

From third figure,

- BH sin (180° + 1) =

sin A. OH

ОВ cos (180° + A) =

OH

- BH
tan (180° + A) = tan A.

ОВ
From fourth figure,

- BH sin (-1) =

sin A. OH

OB cos (- A) =

ОН

- BH tan (-A)

tan A.

OB
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.
I. Any ratio of 2n. 180° + A that same ratio of 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 01.

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