138 Trigonometry; cosine and sine; addition formulae Ch. 9
9.3.4 Subtractionofangles .........................
Definition.Forallα∈
A(F), we denote the an-
gleβin 9.3.3(v) by−α.
Thedifferenceγ−α in
A(F)is defined by spec-
ifying that γ−α=γ+
(−α). In this way we deal
withsubtraction.
P 1
P 2
O
J
I Q
R
S
T
i(α)
−α α H 1
H 2
H 4 H 3
Figure 9.10.
For allα∈A(F),
cos(−α)=cos(co−spα)=cosα, sin(−α)=sin(co−spα)=−sinα.
Proof. WithP 2 as in the proof of 9.3.3(v), we have
cos(−α)=
k−|Q,U|
k
,sin(−α)=
k−|R,V 1 |
k
,
and|R,V 1 |=|T,V|= 2 k−|R,V|. We use this in conjunction with 9.2.1.
9.3.5 Integer multiples of an angle .....................
Definition.Foralln∈Nand allα∈A(F),nαis defined inductively by
1 α=α,
(n+ 1 )α=nα+α, for all n∈N.
We refer tonαasinteger multiplesof the angleα.
For allα∈A(F),
(i) cos( 2 α)=cos^2 α−sin^2 α=2cos^2 α− 1 = 1 −2sin^2 α,
(ii) sin( 2 α)=2cosαsinα.
Proof. These are immediate by 9.3.3 and 9.2.3.
9.3.6 Standard multiples of a right-angle .................
The angles (^90) F, (^180) F, (^270) Fhave the following properties:-