Geometry with Trigonometry

Sec. 9.3 Angles in standard position 135

Ifαandγare different angles inA∗(F),then|α|◦=|γ|◦.
Proof. This is evident ifαandγare both wedge or straight angles and hence, by
addition of 180, if they are both reflex or straight. Ifαis wedge or straight andγis
reflex, then|α|◦≤ 180 ,|γ|◦>180.
NOTATION. Given any real numberxsuch that 0≤x≤360, we denote the angle
α∈A∗(F)with|α|◦=xbyxF. Thus the null, straight and full angles inA∗(F)
are denoted by 0F, 180Fand 360F, respectively.

9.3.2 Addition of angles ...........................

COMMENT. Given anglesα,β∈A∗(F)we wish to define two closely related
forms of addition, the first suited to angle measure as to be dealt with in Chapter 12
and the second suited to more general situations. As we make more use of the latter
we employ for it the common symbol+,and⊕for the former. Asα⊕βis to be an
angle we need to specify its support and its indicator; similarly forα+β.
Definition.Letα,βbe angles inA∗(F)with supports|QOP 1 ,|QOP 2 , respec-
tively. Letlbe the midline of|P 1 OP 2 and letP 3 =sl(Q).Thenα⊕βis an angle with
support|QOP 3 for whichi(α⊕β)⊂H 1 ,sothatα⊕β∈A∗(F). This identifies
α⊕βuniquely except whenP 3 =Q; in this case both the null angle 0Fand the full
angle 360Fhave support|QOQand we defineα⊕βto be this full angle 360Fin
every case except whenαandβare both null; in the latter case we define the sum to
be this null angle 0F. We callα⊕βthesumof the anglesαandβ.

P 1

P 2

P 3

O

J

I Q

R

S

T

α

β

H 1

H 2

H 4 H^3

Figure 9.9. Addition of angles.

For all anglesα,β∈A∗(F),

(i) cos(α⊕β)=cosαcosβ−sinαsinβ,

(ii) sin(α⊕β)=sinαcosβ+cosαsinβ.

Proof. On using the notation of 7.3.1 and above, we have

a 1 =cosα,b 1 =sinα,a 2 =cosβ,b 2 =sinβ,a 3 =cos(α⊕β),b 3 =sin(α⊕β).