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The direction so chosen is called the Positive; that opposite to it is called the Negative.
Again, the addition of any directed magnitude was interpreted to mean the same as the subtraction of the oppositely directed magnitude.
Hence, the algebraical signs of affection exactly answer the purpose of denoting two oppositely directed magnitudes.
It is clearly immaterial which direction we choose to represent by positive numbers and which by negative.
295. The method of applying signs of affection to denote directed angles is the same as that for directed lines.
In Chap. I., it is shown that a trigonometrical angle, corresponding to a given geometrical angle, may be of any magnitude.
When further we distinguish the Initial from the Final line, we can also apply either sign of affection to represent an angle.
In Arts. 14–16, the general value of the magnitude of a trigonometrical angle is given in terms of the corresponding geometrical angle. We may apply these results to a directed trigonometrical angle. For, whichever direction the revolution takes, a complete revolution brings the revolver back to its initial position, and a half-complete revolution brings it to the initial position reversed. Hence the following table gives the value of the trigonometrical directed angle corresponding to any position of the final line :Final line in
Directed angle = 1st Positive Quadrant 2n. 180° + a positive acute angle. 2nd
(2n + 1) 180° 3rd
(2n + 1) 180° + 4th
where n is 0 or any positive or negative integer.
Dependent Signs. 296. We may choose arbitrarily either direction of revolution as positive.
But it is convenient to make the sign for an area depend on that for a revolution. Thus
Let S be any point taken inside a closed area.
Let an elastic string tied to Srevolve round S through four right-angles, and let its other extremity trace out the periphery of the area, so that the string just traverses the whole area.
Then we take the area as positive, when its periphery is traced out in the direction corresponding to the positive direction for the angle.
It should be noted that if S is taken outside the area, the string will have to revolve first through an angle in one direction and then through an equal angle in the opposite direction, in order that its extremity may trace out the periphery. Hence to connect the sign of an area with the sign of an angle we must take S inside the area.
297. When any line is taken independently, either direction along it may be arbitrarily chosen as positive.
But when it is essential to regard a certain line as perpendicular to another line (whose positive direction has already been assigned) the sign of the former depends on the latter.
Thus, the line, which makes a positive right-angle with an independently positive line, is dependently positive: and hence, the line, which makes a negative right-angle with an independently positive line, is dependently negative.
Signs in Multiplication.
298. Suppose ABCD is any rectangle. (See fig. p. 9.)
Then, by an extension of the meaning of multiplication (explained in Art. 21) we have
area ABCD= length AB x length BC.
By a still further extension we may say directed area (ABCDA) =
(AB) x (BC) or (BC) (CD) or (CD) (DA) or (DA) * (AB), directed area (CBADC) =
(CB) (BA) or (BA) * (AD) or (AD) ~ (DC) or (DC) R (CB).
In each of these products the directed factors (and the letters composing those factors) are written in the order indicated by the designation of the directed area.
Let us choose arbitrarily the sign of the first factor. Then that of the second, since it is essential to regard it as perpendicular to the first, will be chosen dependently on the first.
Suppose, then, that (ABCDA) is considered positive, and (C BADC) is considered negative.
I. Let (AB) be arbitrarily chosen as positive, so that (CD) is negative.
Then (BC) makes a positive right-angle with AB, and is, therefore, dependently positive.
But (DA) makes a negative right-angle with AB, and is, therefore, dependently negative.
Thus, the rule of signs in multiplication are applicable to the four equations
(ABC DA) = (AB) (BC) = (CD) (DA),
(CBADC)= (BA) * (AD) = (DC) (CB). II. Let (BC) be arbitrarily chosen as positive ; so that (DA) is negative.
Then (CD) makes a positive right-angle with (BC), and is, therefore, dependently positive.
But (AB) makes a negative right-angle with (BC) and is, therefore, dependently negative.
Thus the rule of signs in multiplication applies to the remaining four equations,
(ABCDA)= (BC) (CD) = (DA) * (AB),
299. The student must carefully distinguish between the multiplication of a directed length by a directed length and the multiplication of a directed length by a number, positive or negative.
Length x Length = Area.
But Length x Number = Length.
Hence multiplying a directed length by a negative number has the effect of altering the length and reversing its direction.
§ 2. APPLICATION TO TRIGONOMETRICAL RATIOS. 300. Let a straight line, terminated at a fixed apex, describe an angle by revolving in a plane from an initial to a final position.
Let a transverse line, perpendicular to either the initial or the final line, cut both these lines (either being produced if necessary). Then
(1) The part cut off from the transversal-directed from the initial to the final line—is called the Perpendicular.
(2) The parts cut off from the initial and final lines—directed from the apex of the angle—are called Base and Hypothenuse (that being hypothenuse which is opposite the right-angle.)
Thus, let OI be the initial, OF the final line.
Then if B is on 01, and BH perpendicular to 01; (OH) is hypothenuse; (OB) is base; and (BH) is perpendicular.
But if B is on OF, and BH perpendicular to OF; (OH) is hypothenuse; (OB) is base; and (HB) is perpendicular.
Having then defined base, hypothenuse, and perpendicular as directed lengths, the ratios of any directed angle whatever are defined in the same way as those of an acute angle were defined in Art. 71–73, except that directed lengths are substituted for mere lengths.
Conventions as to Signs. 301. In order to express the Trigonometrical Ratios algebraically we must affix siyns of affection to the four directed magnitudes in question, viz. :
(1) The angle. (2) The initial line. (3) The final line. (4) The transverse line.
Mathematicians adopt the following conventions in order to express algebraically the ratios of any directed angle :
(1) Lengths along either bounding line of the angle are regarded as positive.
(2) Lengths perpendicular to either bounding line have the same sign as the right-angle which they make with that bounding line.
Thus, the bounding lines being assumed to be positive, the perpendiculars to them receive a dependent sign, as explained in Art. 297.
302. The mathematician thus leaves two directions to be arbitrarily chosen, viz. :
(1) The positive direction of revolution.
303. It is very desirable to fix upon some constant directions of these two, for the purpose of mentally retaining the signs of the different ratios of different angles.
For this purpose we have what may be called a teacher-and student's convention ; viz. :
(1) The positive direction of revolution shall be opposite to the motion of the hands of a clock; i.e. a Right-Up-Left-DownJ. T.