130 Trigonometry; cosine and sine; addition formulae Ch. 9
Definition. Referring to 2.3.3, for each angle support|BACletl=ml(|BAC)
as in 3.6 and 4.1.1. WhenA∈BC, we calll∩IR(|BAC)andl∩ER(|BAC)the
indicatorsof the wedge-angle(|BAC,IR(|BAC))and of the reflex-angle(|BAC,
ER(|BAC)), respectively. WhenA∈[B,C]we calll∩H 1 ,l∩H 2 the indicators of
the straight-angles(|BAC,H 1 ),(|BAC,H 2 ), respectively. The null-angle with sup-
port|BABhas indicator[A,B. The full-angle with support|BABhas indicator[A,D
whereD=AandA∈[B,D]. In each case an indicator is a half-line oflwith initial
point the vertexA. We denote the indicator of an angleαbyi(α).
COMMENT. The first use we make of the concept of indicator is in defining the
cosine and sine of any angle.
9.2 Cosineandsineofanangle ......................
9.2.1 .....................................
Definition. Consider an angleα
with support|BACsuch that the
indicator i(α)lies in a closed
half-plane H 1 with edge AB.
Then we define cosαand sinα
as follows. Take any pointP=A
on[A,C,letQ∈[A,B be such
that|A,Q|=|A,P|andR∈H 1
be such that|A,R|=|A,P|and
AR⊥AB.
P
C
A
U
Q
B
R
V
T
i(α)
H 1Figure 9.3. Cosine and Sine.LetU,Vbe the feet of the perpendiculars fromPtoAB=AQandARrespectively.
Then we define
cosα=|A,P|−|Q,U|
|A,P|
,sinα=|A,P|−|R,V|
|A,P|
.
It follows from this definition that ifH 2 is the other half-plane with edgeABand
if we takeT∈H 2 so that|A,T|=|A,P|andAT⊥AB,then
cos(co−spα)=|A,P|−|Q,U|
|A,P|,sin(co−spα)=|A,P|−|T,V|
|A,P|.
COMMENT. Two comments on this definition are in order. First we note that
whenA,B,Care collinear,H 1 andH 2 are not uniquely determined above but are
interchangeable with each other, so that the anglesαand co-spαare not uniquely
determined. Our second comment is that to show that cosα,sinαare well-defined it
is first necessary to use the ratio results for triangles to show that the values of cosα
and sinαdo not depend on the particular pointP∈[A,Ctaken, and then to show that
if the arms[A,Band[A,Care interchanged the outcome is unchanged.