150 Complex coordinates; sensed angles; angles between lines Ch. 10
(iii) For by (i) and (ii) of the present theorem,cisθ.cis(−θ)=cis(θ−θ)=cis 0F= 1.(iv) For the complex conjugate of cosθ+ısinθ is cosθ−ısinθ=cos(−θ)+
ısin(−θ).
(v) For|cisθ|^2 =cos^2 θ+sin^2 θ= 1.
10.3 Rotations and axial symmetries ....................
10.3.1Rotations................................
Definition.LetZ 0 ∼Fz 0 ,tbe the translationtO,Z 0 ,F′=t(F)andI 0 =t(I).Let
α∈A(F′). The functionrα;Z 0 :Π→Πdefined by
Z∼Fz,Z′∼Fz′,rα;Z 0 (Z)=Z′ifz′−z 0 =(z−z 0 )cisα,is calledrotationabout the pointZ 0 through the angleα.
O IJ H 1
H 2
H 4 H 3
Z 0 I 0
J 0
Z 1
Z
Z′
αθ′θFigure 10.2.If rα;Z 0 (Z)=Z′we have the following properties:-(i)In all cases|Z 0 ,Z′|=|Z 0 ,Z|, and hence in particular rα;Z 0 (Z 0 )=Z 0.(ii)If Z=Z 0 ,θ=∠F′I 0 Z 0 Z andθ′=∠F′I 0 Z 0 Z′,thenθ′=θ+α.(iii)If Z 0 ∼Fz 0 ,Z∼Fz,Z′∼Fz′,thenrα;Z 0 has the real coordinates formx′−x 0 =cosα.(x−x 0 )−sinα.(y−y 0 ),
y′−y 0 =sinα.(x−x 0 )+cosα.(y−y 0 ),