Geometry with Trigonometry

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:-