134 Trigonometry; cosine and sine; addition formulae Ch. 9
P
O
J
I Q
R
S
T
i(α)
H 1
H 2
H 4 H 3
P
O
J
I Q
R
S
T
i(α)
H 1
H 2
H 4 H 3
P
O
J
I Q
R
S
T
i(α)
α H 1
H 2
H 4 H 3
Figure 9.7.
P
O
J
Q
I
R
S
T
i(α)
H 1
H 2
H 4 H 3
Definition. We recall from 3.7 our extension of degree-measure to reflex-angles. Let
|QOPbe a non-straight support, and letH 1 ,H 2 be the closed-half planes with edge
OQ, withP∈H 2 .Letαbe the reflex angle with support|QOP,sothati(α)⊂H 1.
LetSbe the point such thatO=mp(Q,S).Letβbe the wedge or straight angle with
support|SOP.
P
O
J
I Q
R
S
T
i(α)
α
β
H 1
H 2
H 4 H 3
Figure 9.8. Measure of a reflex angle.
Then we defined the degree measure ofαby|α|◦= 180 +|β|◦. In particular if
P=Q,thenβis a straight angle,αis the full angle with support|POP=|QOQand
indicator[O,S,and|α|◦= 360.
Definition. Given a frame of referenceF=([O,I,[O,J), we denote byA∗(F)
the set of anglesαwith arm[O,I and with indicatori(α)⊂H 1. These angles are
saidtobeinstandard positionwith respect toF.