240 Trigonometric functions in calculus Ch. 12
Definition. Given any angleα∈A∗(F), in 9.4.1 we defined^12 α.Nowforany
integern≥ 1 , 21 nαis defined inductively by
1
21
α=
1
2
α, and for all n≥ 1 ,
1
2 n+^1
α=
1
2
[
1
2 n
α].
For allα∈A∗(F),|^12 α|◦=^12 |α|◦.
Proof. We use the notation of 9.4.1. From the definition of midlQOPthis is im-
mediate whenP∈H 1 , so we suppose thatP∈H 2 ,P=S,P=Q.Then
|α|◦= 180 +|∠SOP|◦=|∠QOP′|◦+|∠P′OS|◦+|∠SOP|◦
=|∠QOP′|◦+|∠P′OP|◦= 2 |∠QOP′|◦
by 3.7.1.
For this, note that[P,P′]meets
OQ in a point G.IfH =
mp(Q,P),thenP′∈OH and
ifH 5 ,H 6 are the closed half-
planes with common edgeOH,
withQ∈H 5 ,thenasH∈[Q,P]
we haveP∈H 6 .Then[P′,P]⊂
H 6 so G∈H 6 .But[O,Q⊂
H 5 ,[O,S⊂H 6 soG∈[O,S.
Thus[O,S⊂IR(|P′OP).
P
P′
G
H
O
J
I Q
R
S
T
i(α)
α H 1
H 2
H 4 H 3
H 5
H 6
Figure 12.1.
Ifα,β∈A∗(F)are such that
|α|◦+|β|◦≤ 360 ,then
1
2 (α⊕β)=
1
2 α⊕
1
2 β.
Proof. By the commutativity
and associativity of⊕,
(^12 α⊕^12 β)⊕(^12 α⊕^12 β)
=(^12 α⊕^12 α)⊕(^12 β⊕^12 β)
=α⊕β,
so we show that under the con-
dition|α|◦+|β|◦≤360, if^12 α⊕
1
2 βhas support|QOP
′ 3 thenP 3 ′∈
H 1.