Geometry with Trigonometry

Sec. 10.2 Complex-valued distance 149

O I

J H^1

H 2

H 4 H 3

Z 0 I 0

J 0 Z

θ

Figure 10.1.

O I

J H^1

H 2

H 4 H 3

Z 0 I 0

J (^0) Z 1 Z
α
IfZ 1 =Z 0 ,Z 1 ∼Fz 1 andα=∠F′I 0 Z 0 Z 1 ,thenbythis
x 1 −x 0 =kcosα,y 1 −y 0 =ksinα,
wherek=|Z 0 ,Z 1 |. On inserting this in 6.3.1 Corollary, we see that
Z 0 Z 1 ={Z≡(x,y):(x−x 0 )sinα−(y−y 0 )cosα= 0 }.
WhenZ 0 Z 1 is not parallel toOJwe have that cosα=0 and this equation of the line
Z 0 Z 1 can be re-written asy−y 0 =tanα(x−x 0 )where tanα=sinα/cosα. We call
tanαtheslopeof this line.
Notation. For any angleθwe write cisθ=cosθ+ısinθ.
The complex-valued functioncishas the properties:-
(i)For allθ,φ∈A(F),cis(θ+φ)=cisθ.cisφ.
(ii) cis 0F= 1.
(iii) For allθ∈A(F),cis^1 θ=cis(−θ).
(iv) For allθ∈A(F),cisθ=cis(−θ),where ̄z denotes the complex conjugate of
z.
(v)For allθ,|cisθ|= 1.
Proof.
(i) For
cisθ.cisφ=(cosθ+ısinθ)(cosφ+ısinφ)
=cosθcosφ−sinθsinφ+ı[sinθcosφ+cosθsinφ]
=cos(θ+φ)+ısin(θ+φ)=cis(θ+φ).
(ii) For cis 0F=cos0F+ısin0F= 1 +ı 0 = 1.