172 Complex coordinates; sensed angles; angles between lines Ch. 10
is equal to
k^2 [(x 5 −x 4 )(y 5 −y 4 )+x 5 y 4 +x 4 y 5 ]−(x 5 y 4 +x 4 y 5 )(x 4 x 5 +y 4 y 5 )
=k^2 (x 5 y 5 +x 4 y 4 )−[x 4 y 4 (x^25 +y^25 )+x 5 y 5 (x^24 +y^24 )]
=(k^2 −k^2 )(x 5 y 5 +x 4 y 4 )= 0.
Similarly the numerator in
1
2
(y 5 −y 4 )^2 −(x 5 −x 4 )^2
k^2 −(x 4 x 5 +y 4 y 5 )
−x^4 x^5 −y^4 y^5
k^2
equals
k^2 [(y 5 −y 4 )^2 −(x 5 −x 4 )^2 − 2 (x 4 x 5 −y 4 y 5 )]+ 2 (x 4 x 5 +y 4 y 5 )(x 4 x 5 −y 4 y 5 )
=k^2 [y^25 +y^24 −x^25 −x^24 ]+ 2 (x^24 x^25 −y^24 y^25 )
=k^2 [y^25 +y^24 −x^25 −x^24 ]+ 2 [x^24 (k^2 −y^25 )−y^24 y^25 ]
=k^2 [y^25 +y^24 −x^25 −x^24 ]+ 2 [x^24 k^2 −y^25 (x^24 +y^24 )]
=k^2 [y^25 +y^24 −x^25 −x^24 + 2 x^24 − 2 y^25 ]= 0.
To apply these we note that by 10.10.4
sinαd=
y 4
k
,cosαd=
x 4
k
,sinβd=
y 5
k
,cosβd=
x 5
k
.
The sumγd=αd+βdhas side-linesOQandOZ 6 , and we sub-divide into two major
cases. First we suppose thatx 5 y 4 +x 4 y 5 >0 or equivalently|αd|◦+|βd|◦<180. Then
y 6 >0 and we have
sinγd=y^6
k
,cosγd=x^6
k
.
It follows from (10.10.2) that
sin(αd+βd)=sinαdcosβd+cosαdsinβd,
cos(αd+βd)=cosαdcosβd−sinαdsinβd.
Secondly we suppose thatx 5 y 4 +x 4 y 5 <0 or equivalently|αd|◦+|βd|◦>180.
Theny 6 <0sowehave
sinγd=−
y 6
k,cosγd=−
x 6
k.
It follows from (10.10.2) that
−sin(αd+βd)=sinαdcosβd+cosαdsinβd,
−cos(αd+βd)=cosαdcosβd−sinαdsinβd.
There is a further case whenx 5 y 4 +x 4 y 5 =0 and we obtain these formulae according
asx 4 x 5 −y 4 y 5 is positive or negative, respectively. Thus the addition formulae for sine
and cosine of duo-angles are more complicated than those of angles.