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5.6 The Law of Sines
How do we solve triangles that are not
right triangles?
B
a
c
A
C
b
Dec 21­2:20 PM
B
a
c
A
b
C
The Law of Sines: Let ΔABC be any triangle
with a, b and c representing the measures of
the sides opposite the angles with measures A,
B and C, respectively. Then, the following is true.
a =
sinA
b
sinB
=
c
sinC
Jan 3­3:17 PM
1
Examples: Solve each triangle. Round to
the nearest tenth.
1.) B = 100o , C = 50o , c = 30
Dec 21­2:30 PM
2.) c = 19.3 , A = 39o15' , C = 64o45'
Dec 21­2:36 PM
2
Area of Triangles
Let ΔABC be any triangle. Then the area K
can be determined by using one of the following
formulas:
K = ½bc sinA
K = ½ab sinC
K = ½ac sinB
Note: Use these when given
two sides and one angle.
K = ½a2sinB sinC
sinA
K = ½c2 sinA sinB
sinC
K = ½b2 sinA sinC
sinB
Note: Use these when given
one side and two angles.
Dec 21­2:43 PM
Examples: Find the area of each triangle.
Round to the nearest tenth.
1.) a = 5, B = 37o , C = 84o
2.) b = 146.2, c = 209.3 , A = 62.2o
Dec 21­2:50 PM
3
Homework Hint!
Problem #31, page 317: A hot air balloon is flying
above Groveburg. To the left side of the balloon,
the balloonist measures the angle of depression to
the Groveburg soccer fields to be 20o15'. To the
right side of the balloon, the balloonist measures
the angle of depression to the high school football
field to be 62o30'. The distance between the two
athletic complexes is 4 miles.
Set up the picture!
a
b
h
x
4-x
Jan 3­2:57 PM
5.6 ICE
You may work together, but each person must submit their
own work.
1.) Given A = 65o , B = 50o and c = 12:
a.) Solve the triangle. Round to the nearest tenth.
b.) Find the area of the triangle. Round to the nearest tenth.
2.) Challenge (Bonus): Derive the Law of Sines from the
B
following figure:
c
A
a
h
C
b
Dec 21­2:55 PM
4