Geometry with Trigonometry

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.