Geometry with Trigonometry

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.