Geometry with Trigonometry

Sec. 9.5 The cosine and sine rules 141

A

B

C

D

Figure 9.12.

A

B

C D

A

B

D C

The cosine rule.In each triangle[A,B,C],

cosα=

b^2 +c^2 −a^2

2 bc

,cosβ=

c^2 +a^2 −b^2

2 ca

,cosγ=

a^2 +b^2 −c^2

2 ab

ProofOn returning to the last proof, we note that whenD∈[B,Cwe have

cosβ=

|B,D|

|B,A|

=

x

c

,

while

x=

c^2 +a^2 −b^2

2 a

,

and so

cosβ=

c^2 +a^2 −b^2

2 ca

Similarly, whenB∈[D,C]we have

cosβ=−

|B,D|

|B,A|

=−

x

c

,

while

x=−c

(^2) +a (^2) −b 2
2 a

,

and this gives the same conclusion.

9.5.2 Thesinerule..............................

The sine ruleIn each triangle[A,B,C],

sinα

a

=

sinβ

b

=

sinγ

c

.

Proof. By the cosine rule

cos^2 α

a^2

=

1 −sin^2 α

a^2

=

(b^2 +c^2 −a^2 )^2

4 a^2 b^2 c^2

,