Geometry with Trigonometry

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.