Objectives:
1. To find the area of a
triangle given 2 sides
and the included
angle
2. To derive and use
the Law of Sines
•
•
•
•
•
Assignment:
P. 436: 1-18 S
P. 436: 19-24 S
P. 436: 25
P. 436: 29-34 S
P. 436-7: 35-43 S
You will be able to find the area of a triangle
given 2 sides and the included angle
Find the area of ΔABC.
The area of an oblique triangle is half the product of
two sides and the sine of the included angle.
1
A  bc sin A
2
1
A  ac sin B
2
1
A  ab sin C
2
Find the area of a triangular lot containing side
lengths that measure 24 yards and 18 yards
which form a 80° angle.
You will be able to derive and use the Law of
Sines
As the previous exercises illustrate, trigonometry
can be applied to non-right or oblique
triangles. In the first exercise, we used it to
find an unknown height. Now, we’ll use it to
find a missing side length of a non-right
triangle.
Step 1: Draw ΔABC as
shown with height h.
Notice that side a is
opposite <A, b is
opposite <B, and c is
opposite <C.
Step 2: Write an
equation for h using
sin(<A).
Step 3: Write an
equation for h using
sin(<B).
Step 4: Use
substitution to
combine the two
previous equations
and write your final
equation as a
proportion.
Step 5: Write an
equation for k using
sin(<B).
Step 6: Write an
equation for k using
sin(<C).
Step 7: Use
substitution to
combine the two
previous equations
and write your final
equation as a
proportion.
Step 8: Finally, use the
transitive property to
combine the
proportion from S4
and S7. This is the
Law of Sines.
If ΔABC has side lengths a, b, and c as shown,
then
sin A sin B sin C


.
a
b
c
Use to find a missing angle
If ΔABC has side lengths a, b, and c as shown,
then
a
b
c


.
sin A sin B sin C
Use to find a missing side
Solve ΔABC.
Solve ΔABC.
The Law of Sines can also be used to find a
missing angle measure, but only sometimes.
The geometric reason is that SSA is not a
congruence shortcut.
C
160
C
260
260
160
36
B1
36
A
B2
A
Show that there is no triangle for which A = 60°,
a = 4, and b = 14.
Solve ΔABC.
Find two triangles for which A = 58°, a = 4.5, and
b = 5.
When using the Law of Sines to
find a missing angle measure
(SSA), you could have 0, 1, or 2
possible answers. Just set up
your problem, and solve as
usual. If you do get an angle
measure that works, try
subtracting it from 180°. If it
also works, you’ve got 2
answers!
Find the indicated measure.
1. x and y
2.  and 
Find the value of β.
Find the length of AC in acute triangle ABC.
As the previous example
demonstrates, you
cannot always use the
Law of Sines for every
triangle. You need either
two sides and an angle or
two angles and a side in
the following
configurations:
AAS
SSA
• Ambiguous
You’re an avid swimmer, and when you see a
lake, you just have to swim in it. You start at
one point on the south side of the lake and
swim at a bearing of 028. Then you swim to a
point on the north side of the lake at a bearing
of 302. Finally, you swim the 800 m back to
your starting point, which is coincidentally due
south. How many total meters did you swim?
Objectives:
1. To find the area of a
triangle given 2 sides
and the included
angle
2. To derive and use
the Law of Sines
•
•
•
•
•
Assignment:
P. 436: 1-18 S
P. 436: 19-24 S
P. 436: 25
P. 436: 29-34 S
P. 436-7: 35-43 S