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But since (OP) is directed the same way as OF, the final line, we have, (representing OP algebraically),

(OP) = + arithmetic distance between 0 and P. Now at the limit, (MP) vanishes, and (OM) coincides with (OP). Thus

sin 180o = 0 and cos 180o =-1. This last result is deserving of special attention.

In representing (OM) by an algebraical quantity we refer to (01) in order to determine its sign: but in representing (OP) we refer to (OF).

In the limit (OF) becomes opposite to (OI); and hence the signs of (OM) and (OP) are opposite; although they exactly coincide in direction as well as magnitude.

This exemplifies the principle that lines which coincide (so to speak) purely incidentally have not necessarily the same sign.

Consider the angle 270o.

Here (OM) vanishes, and (MP) ultimately coincides with (OP).

But (MP) makes a negative right-angle with the initial line (OI), and is therefore represented algebraically by a negative number.

On the other hand, (OP) which is along the final line is represented by a positive number. Hence

sin 270° - 1 and cos 270o = 0.

316. The principle of continuity confirms these conclusions.

For, when the final line is in the 2nd or 3rd quadrant, the cosine is negative. Hence, when it separates these quadrants, , viz. when the angle is (2n + 1) 180°, its cosine must be negative.

Again, when the final line is in the 3rd or 4th quadrant, the sine is negative. Hence, when it separates these quadrants, viz. when the angle is (4n – 1) 90°, its sine must be negative.

o to 0.

317. To trace the changes in magnitude and sign of the ratios of an angle as it increases from 0 to 360°.

As an example, take the tangent.
From 0° to 90°, tangent is + ve, and increases from 0 to +.co.
From 90° to 180°, tangent is – ve, and increases from
From 180° to 270°, tangent is + ve, and increases from 0 to +0.
From 270° to 360°, tangent is – ve, and increases from

0 to 0. It should be observed that the ratios which become infinite at the limiting values 0°, 90°, 180°, 270° are ambiguous in sign : i.e. their sign depends upon whether we arrive at the limit by increasing the angle up to its limit or by decreasing the angle down to its limit.

318. To examine how the fundamental formula connecting the ratios of an angle are affected by taking the angle of any magnitude or sign.

The formulæ of Art. 79, which follow immediately from definition, are evidently unaffected by extending the angle to any magnitude or sign.

Also the formulæ of Art. 80, which involve the squares of the ratios, are unaffected by the signs of the ratios.

But, in applying these latter to express a ratio in terni of another by taking the square root, we must observe that

sin A == (1 - cos” A); cox A = + (1 + cot? A);
tan A = + (seco A – 1); cot A = + (coxo A – 1);
sec A

= + (1 + tano A); cos A === (1 - sin? A).
The upper sign must be taken
in the 1st line, for angles of the 1st or 2nd quadrant;
in the 2nd line, for angles of the 1st or 3rd quadrant;
in the 3rd line, for angles of the 1st or 4th quadrant.

319. This result shows that, knowing one ratio of an angle and nothing more, there are two possible equal and opposite values for any other ratio (except the reciprocal of that given).

This is geometrically obvious.

For example; knowing that the cotangent of an angle has some particular value, which is (say) negative, the angle may be in the 2nd or 4th quadrant; and, therefore, either sine, cosine, secant, or cosecant may be positive or negative.

§ 3. INVERSE SYMBOLS. 320. The symbols sin-, cos- &c. are used in the following

sense.

If & be any angle, whose sine has some given value a, then for 8 we may write sin-1 a.

And so for the other ratios.
Thus the equations

sin A = a and 0 = sin-1 are precisely equivalent statements.

The latter is obtained from the former by symbolically dividing' both sides by the function 'sin', as if sin were an algebraic factor.

In working out problems involving these (so called) inverse symbols, it is only necessary to equate each inverse symbol to some new letter, and translate the resulting equation into the ordinary form by means of the equivalence above explained.

This method is similar to that required in logarithms.
Example. To show that
tan-1a + tan-1b = tan-1

1 - ab
Let tan-1 a = 0, this means a tan .
Let tan-1 b = 0, this means b= tan .

tan 0 + tan Now

tan (0+°)

a +6

1- tan 0 tan •

a + b

i.e.

1 - ab Translating this last equation back into the inverse notation, 0 + , i.e. tan-1a + tan-16, = tan-1

1 + ab

a + b

321. The following points must be carefully observed :

(1) Whereas sin 0 is a numerical quantity = a (say); sin- a is an angle.

(2) Whereas, when 6 is given, sin has only one value = a (say); when a is given, sin- a has an indefinite number of values.

In fact the results of Art. 313, expressed in inverse notation, become as follows:

General value of sin-1 a = n. 180° + (-1)" (some particular value of sin-1 a]

General value of tan-la=n. 180° + [some particular value of tan-a]

General value of cos-?a=n. 360° + [some particular value of cos-a] where n has any integral value whatever.

S 4.

ADAPTATION OF RESULTS TO A TRIANGLE.

322. In applying the theory of directed magnitudes to the formulæ of a triangle, it must be noticed that we have three primary lines to consider.

Take the triangle ABC.

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Consider the results obtained by dropping AL perpendicular upon BC. We have

(BL)
cos (CBA) = and sin (CBA) =

(LA)
(BA)

(BA)

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=

:: (BA) cos (CBA) = (BL) and (BA) sin (CBA) = (LA). Also (CL) (LC)

(AL) (LA) cos (ACB) =

and sin (ACB) (CA) (AC)

(CA) (AC) :: (AC) cos (ACB)= (LC) and (AC) sin (ACB) =(LA). Comparing these results, since (BC) = (BL) + (LC),

(BA) cos (CBA)+(AC) cos (ACB)= (BC),

(BA) sin (CBA)= (AC) sin (ACB). Also the triangle (ABCA) = } (BC) (LA).

323. Now let us take the area (ABC A) to be positive; so as to correspond to the positive rotation from (BC) to (BA), i.e., the angle (CBA); and to that from (AC) produced to (BC) produced, i.e. the angle (ACB).

Then take (BC) as an initial line, which will, therefore, be primarily positive.

Hence (LA), which makes a positive right-angle with (BC), is secondarily positive.

Of the angle (CBA), (BC) is the initial, (BA) is the final line; these are, therefore, positive.

Of the angle (ACB), (CA) is the initial line reversed, (CB) is the final line reversed; these are, therefore, negative.

(LC) is positive or negative according as it is like or unlike (BC); i.e. according as C is acute or obtuse.

(BL) is positive or negative according as it is like or unlike (BC); i.e. according as B is acute or obtuse.

The signs in all the above equations are, therefore, accounted for.

324. We have, therefore, three separate sets of conventions according to the side and its perpendicular which we consider.

In each case, however, we maintain the direction (ABC A) or (BCAB) or (CA BC) as positive for the area.

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