Advanced High-School Mathematics

24 CHAPTER 1 Advanced Euclidean Geometry

Law of Sines. Given triangle
4 ABCand sidesa, b,andc, as in-
dicated, we have

sinA

a

=

sinB

b

=

sinC

c

.

Proof. We note that

1

2

bcsinA= area 4 ABC=

1

2

basinC,

and so

sinA

a

=

sinC

c

.

A similar argument shows that
sinB
b

is also equal to the above.

Law of Cosines. Given triangle
4 ABCand sidesa, b,andc, as in-
dicated, we have

c^2 =a^2 +b^2 − 2 abcosC.