132 Trigonometry; cosine and sine; addition formulae Ch. 9
When insteadP∈H 4 ,wehaveO∈[Q,U]so|O,U|=|Q,U|−kand similarly
|O,U 1 |=|Q 1 ,U 1 |−k 1. On inserting these we obtain
k−|Q,U|
k
=
k 1 −|Q 1 ,U 1 |
k 1
again.
WhenP∈OIwe have eitherP=QorP=S.WhenP=Q,wehaveP 1 =Q 1 and
the formula checks out. It checks out similarly in the cases whenPisR,SorT.
By a similar proof we find that
k−|R,V|
k
=
k 1 −|R 1 ,V 1 |
k 1
Thus it makes no difference to the values of cosαand sinαifPis replaced byP 1.
P
P 1
O
J
U 1 I U Q
R
S
V
V 1
T
i(α)
H 1
H 2
H 4 H 3
Figure 9.5.
P
O
J
U
X
sl(H 1 )
Q
W
I
R
S
V
Y
T
i(α)
H 1
H 2
H 4 H 3
Figure 9.6.
(iii) It remains to show that if the arms[A,Band[A,Care interchanged the out-
come is unchanged. Letl=midlQOPso thatsl(OQ)=OPandsl(H 1 )is a closed
half-plane with edgeOP.Asi(α)⊂H 1 we havesl(i(α))⊂sl(H 1 );butasi(α)⊂l,
sl(i(α)) =i(α)and thusi(α)⊂sl(H 1 ).IfW=sl(R)thenW∈sl(H 1 )and asOQ⊥
ORwe haveOP⊥OW. MoreoverX=sl(U)=πOP(Q)andY=sl(V)=πOW(Q)
satisfy|P,X|=|Q,U|,|W,Y|=|R,V|. Hence
k−|P,X|
k
=
k−|Q,U|
k
,
k−|W,Y|
k
=
k−|R,V|
k
This completes the proof.
9.2.2 Polarcoordinates ...........................
For Z=O, let k=|O,Z|and the angleαhave support|IOZand indicator i(α)in
H 1 .ThenifZ≡F(x,y),
x=kcosα,y=ksinα.