Law of Sines

1
Law of Sines

• Let ABC be any triangle with a, b, and c
  representing the measures of sides opposite
  angles with measures A, B, and C, respectively.
  Then
• sin A sin B sin C
• a b c

C
b
a
A
B
c
2
Proof of Law of Sines

• Let ABC be a triangle with an altitude from c
  that
• intersects side AB at D. Let h represent the
  measure of
• CD. Since triangle ACD and BCD are right
  triangles,
• We can find sin A and sin B as follows

sin A h / b sin B h/a Thus, b sin A
h a sin B h b sin A a sin B
C
a
b
h
sin A sin B a b
A
B
D
3
Solve the triangle

• The Law of Sines can be used to solve a
• triangle when you know the
• You know the measures of two angles and any side
  of a triangle.
• You know the measures of two sides and an angle
  opposite one of these sides of the triangle.