136 Trigonometry; cosine and sine; addition formulae Ch. 9
We note that in 7.3.1
(a 1 +a 2 )^2 +(b 1 +b 2 )^2 = 2 ( 1 +a 1 a 2 +b 1 b 2 ),asa^21 +b^21 =a^22 +b^22 =1. Then, by 7.3.2, whenP 1 andP 2 are not diametrically oppo-
site,
cos(α⊕β)−cosαcosβ+sinαsinβ=(a 1 +a 2 )^2 −(b 1 +b 2 )^2 + 2 (−a 1 a 2 +b 1 b 2 )( 1 +a 1 a 2 +b 1 b 2 )
2 ( 1 +a 1 a 2 +b 1 b 2 ),
and the numerator here is equal to
a^21 +a^22 −b^21 −b^22 − 2 a^21 a^22 + 2 b^21 b^22 = 2 a^21 + 2 a^22 − 2 − 2 a^21 a^22 + 2 ( 1 −a^21 )( 1 −a^22 )= 0.
Similarly
sin(α⊕β)−sinαcosβ−cosαsinβ=2 (a 1 +a 2 )(b 1 +b 2 )− 2 (a 1 b 2 +a 2 b 1 )( 1 +a 1 a 2 +b 1 b 2 )
2 ( 1 +a 1 a 2 +b 1 b 2 ) ,and the numerator here is equal to twice
a 1 b 1 +a 2 b 2 −a 1 b 1 (a^22 +b^22 )−a 2 b 2 (a^21 +b^21 )= 0.
WhenP 1 andP 2 are diametrically opposite,
cos(α⊕β)−cosαcosβ+sinαsinβ=b^21 −a^21 −a 1 (−a 1 )+b 1 (−b 1 )= 0 ,
sin(α⊕β)−sinαcosβ−cosαsinβ=− 2 a 1 b 1 −b 1 (−a 1 )−a 1 (−b 1 )= 0.9.3.3 Modified addition of angles ......................
COMMENT. In 9.3.2 we clearly exercised a choice in specifying whatα⊕βshould
be whenP 3 =Q. The choice made there is what suits length of a circle and area of a
disk which will be treated in Chapter 12, and that was the reason for the choice made.
We now define modified additionα+βof angles, which is easier to use.
Definition.LetA(F)=A∗(F){ (^360) F},sothatA(F)is the set of all non-
full angles inA∗(F). We denote by∠FQOP=∠FIOPthe unique angle inA(F)
with support|QOP=|IOP.
Definition.Letα,βbe angles inA(F)with supports|QOP 1 ,|QOP 2 .Letlbe
the midline of|P 1 OP 2 and letP 3 =sl(Q).Thenα+βis the angle inA(F)with
support|QOP 3. Note that whenP 3 =Qwe haveα+β= (^0) F. We callα+βthe
modified sumof the anglesαandβ.
For allα,β∈A(F),
cos(α+β)=cosαcosβ−sinαsinβ,sin(α+β)=sinαcosβ+cosαsinβ.
Proof. This follows immediately from 9.3.2 as cos360F=cos0F,sin360F=
sin0F.