Geometry with Trigonometry

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.