170 Complex coordinates; sensed angles; angles between lines Ch. 10
We extend our frame of referenceFby taking in connection with the line pair
{OI,OJ}a canonical pair of duo-sectorsD 1 andD 2 , withD 1 the union of the first
and third quadrantsQ 1 andQ 3 ,andD 2 the union of the second and fourth quadrants.
For any linelthrough the originO, we consider the duo-angleαdwith side-lines
OIandl, such that the indicatormofαdlies in the duo-sectorD 1 , that is the bisector
of the line-pair{OI,l}which lies in the duo-sector ofαdalso lies inD 1. We denote
byDA(F)the set of such duo-angles, and we say that they are instandard position
respect toF.
Ifl=OIandZ 4 ≡(x 4 ,y 4 )is a point other thanOonl, then so is the point with
coordinates(−x 4 ,−y 4 ); thus, without loss of generality, we may assume thaty 4 > 0
in identifyinglasOZ 4 .ThenZ 4 ∈H 1 and
|αd|◦=|∠IOZ 4 |◦,
cosαd=cos(∠IOZ 4 )=
x 4
√
x^24 +y^24
,
sinαd=sin(∠IOZ 4 )=
y 4
√
x^24 +y^24
.
Whenαdis not a right duo-angle, we have
tanαd=
y 4
x 4
.
We identifyl=OIasOZ 4 whereZ 4 ≡(x 4 , 0 )andx 4 >0. Thus for the null duo-
angle in standard position we have cosαd= 1 ,sinαd= 0 ,tanαd=0. We denote this
null duo-angle by 0dFand the right duo-angle in standard position by 90dF.
We now note that ifαd,βd∈DA(F)andtanαd=tanβd,thenαd=βd.
Proof. For this we letαd,βdhave pairs of side-lines(OI,OZ 4 ),(OI,OZ 5 ), respec-
tively, where|O,Z 4 |=|O,Z 5 |=k, and eithery 4 >0orx 4 > 0 ,y 4 =0, and similarly
eithery 5 >0orx 5 > 0 ,y 5 =0. Then neitherαdnorβdis 90dFand
y 4
x 4
=
y 5
x 5
,x^24 +y^24 =x^25 +y^25 =k^2.
Ify 4 =0theny 5 =0 and both duo-angles are null. Suppose then thaty 4 =0so
thaty 4 >0; it follows thaty 5 >0. Then
k^2 =x^25 +y^25 =
y^25
y^24
x^24 +y^25 =
y^25
y^24
(x^24 +y^24 )=
y^25
y^24
k^2.
Hencey^25 =y^24 ,andsoy 5 =y 4. It follows thatx 5 =x 4.