Geometry with Trigonometry

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.

W

P 1 ′

P 2 ′

P 3 ′=N

P 1

P 2 P^3

O

J

I Q

R

S

T

H 1

H 2

H 4 H 3

Figure 12.2.

Let[O,P 1 ′=i(α),[O,P 2 ′=i(β)with|O,P′ 1 |=|O,P′ 2 |=|O,Q|=k.Then^12 αand

1

2 βare the angles inA

∗

Fwith supports|QOP

′

1 ,|QOP

′

2 , respectively.