Sec. 9.2 Cosine and sine of an angle 131
To help us in our study of angles, it is convenient to fit a frame of reference to
the situation in the definition. We takeO=A,I=BandJ=Oa point inH 1 so that
OI⊥OJ.WeletH 3 ,H 4 be the closed half-planes with edgeOJ, withI∈H 3.
Withk=|A,P|=|O,P|,letQ
be the point on[O,I=[A,B
such that|O,Q|=k,andletR
be the point on[O,Jsuch that
|O,R|=k. ChooseS,Tso thatO
=mp(Q,S),O=mp(R,T).
P
C
A=O
J
B=IU Q
R
S
V
T
i(α)
H 1
H 2H 4 H 3
Figure 9.4.The cosine and sine of an angle are well-defined.
Proof.
(i) WhenA,BandCare collinear there are two cases to be considered. One case is
whenA∈[B,C]so that|BACis straight. Then each ofα,co−spαis a straight-angle
and asP=S,wehaveU=S,V=Aand so
cosα=cos(co−spα)=− 1 ,sinα=sin(co−spα)= 0.A second case is whenC∈[A,Bso that one ofα,co−spαis a null-angle with
indicator[A,Band the other is a full-angle with indicator[A,B 1 whereAis betweenB
andB 1. Both of the indicators are inH 1 andH 2 ,butasP=Qwe haveU=Q,V=A
and so
cosα=cos(co−spα)= 1 ,sinα=sin(co−spα)= 0.
Thus in neither case does the ambiguity affect the outcome.
(ii) We now use the ratio results for triangles to show that the values of cosαand
sinαdo not depend on the particular pointP∈[A,Cchosen. Takek 1 >0andlet
P 1 ,Q 1 ,R 1 be the points in[O,P,[O,Q,[O,R, respectively, each at a distancek 1
fromO.LetU 1 =πOI(P 1 ),V 1 =πOJ(P 1 ).
Suppose first thatP∈OI,P∈OJ.AsPU‖P 1 U 1 , by 5.3.1
|O,U|
|O,U 1 |=|O,P|
|O,P 1 |,
and so
|O,U|
k
=
|O,U 1 |
k 1.
Now ifP∈H 3 so thatU∈[Q,O]and so|O,U|=k−|Q,U|, by 2.2.3(iv)P 1 ∈H 3
and similarly|O,U 1 |=k 1 −|Q 1 ,U 1 |. On inserting this we get that
k−|Q,U|
k=
k 1 −|Q 1 ,U 1 |
k 1