Sec. 9.6 Cosine and sine of angles equal in magnitude 143
Proof. By the last result we haved 2 =c^2 +caacos^12 β,d 3 =a^2 +abbcos^12 γ.Then
d^22 −d 32 = 4 a^2
[
c^2
(c+a)^2
cos^2
1
2
β−
b^2
(a+b)^2
cos^2
1
2
γ
]
= 2 a^2
[
c^2
(c+a)^2
( 1 +cosβ)−
b^2
(a+b)^2
( 1 +cosγ)
]
= 2 a^2
[
c^2
(c+a)^2
(
1 +
c^2 +a^2 −b^2
2 ca
)
−
b^2
(a+b)^2
(
1 +
a^2 +b^2 −c^2
2 ab
)]
=a
[
c−b+
bc^2
(a+b)^2
−
b^2 c
(c+a)^2
]
=a
[
c−b+
bc
(a+b)^2 (c+a)^2
[c(c+a)^2 −b(a+b)^2 ]
]
=a(c−b)
[
1 +
bc
(a+b)^2 (c+a)^2
[a^2 +b^2 +c^2 + 2 ab+bc+ 2 ca]
]
Thenb<cimplies thatd 2 >d 3.
9.6 COSINE AND SINE OF ANGLES EQUAL IN MAG-
NITUDE
9.6.1 .....................................
If anglesα,βare such that|α|◦=|β|◦,thencosα=cosβandsinα=sinβ. Con-
versely ifcosα=cosβandsinα=sinβ,then|α|◦=|β|◦unless one of them is null
and the other is full.
Proof.LetF=([O,I,[O,J)andαhave support|QOPand indicator inH 1 ,
where|O,P|=|O,Q|=k.LetF′=([O′,I′,[O′,J′)andβhave support|Q′O′P′
and indicator inH 1 ′,where|O′,P′|=|O′,Q′|=k.
O I
J
P
Q
H 1
H 2
H 4 H 3
O′
J I′
′
P′
Q′
Figure 9.14.