Download 6 8 2 6 8 cos32 b = + − sin sin 32 6 4.3 A =

Section 6.3: The Law Of Cosines
§1 Difference with Law of Sines
When we want to solve oblique triangles, we use the Law of Sines when we are given two angles and any side,
or if we are given two sides and a non-included angle. We use the Law of Cosines when we are given two sides
and the included angle. We can also use the Law of Sines when are are given all three sides and no angles. Thus
we use Law if Sines for SAA or SSA triangles, and the Law of Cosines with SAS or SSS triangles.
To solve an SAS triangle, use the Law of Cosines to find the side opposite the given angle. Then use the Law of
Sines to find the angle opposite the shorter of the given two sides. Find the third angle by subtracting the two
angles from 180.
To solve an SSS triangle, use the Law of Cosines to find the angle opposite the longest side. Then use the Law of
Sines to find either of the two remaining angles. Then find the third angle by subtracting the two angles from
180.
Example: Solve the following triangle:
This is an SAS triangle. First use the Law of Cosines to find the side opposite the given angle. So we need to find
2
2
2

side b. We end up with b = 6 + 8 − 2 ( 6 )( 8 ) cos 32 . The decimal answer is 4.3. Next we use the Law of Sines
sin A sin 32
to find the angle opposite the shorter side. The shorter side is a. Hence we end up with
. We
=
6
4.3
end up with A = 48 degrees. Note that we don’t need to consider 180-48= 132 degrees as a possibility. Finally,
we find the measure of angle C by subtracting the two known angles from 180 to get 100 degrees.
Example: Solve the following triangle:
This is an SSS triangle. Use the Law of Cosines to find the angle opposite the longest side. The longest side in this
2
2
2
case is c. Hence we use 16 = 10 + 12 − 2 (10 )(12 ) cos C . The answer is C = 93 . Now use the Law of Sines
sin A sin 93
. We end up with
=
10
16
A = 39 . Now we can find B by subtracting A and C from 180. Hence B = 48
to find the angle either A or B (it doesn’t matter). Let’s find A. We can use
PRACTICE
1) Solve the following triangle: a = 6, c = 5, B = 50 degrees
§2 Applications
PRACTICE
2) Two cars leave an intersection at the same time traveling on different roads. The angle of the road between
them makes an angle of 60 degrees. The first car travels 40 miles per hour and the second car travels 60 miles
per hour. How far apart are the cars after one hour?
3) Two ships leave a port at the same time. One travels at 20 miles per hour in a direction N 35 E , and the
other travels at 30 miles per hour in a direction S 25 E . How far apart are the ships after one hour?
§3 Area of a Triangle
We can use trigonometric functions to find the area of certain triangles. From geometry, you should be familiar
with the formula for the area of a triangle. Its A =
1
bh , where b is the base of the triangle and h is the height
2
of the triangle. For example, look at the following figure.
§4 SSS Triangles
Another triangle is called the SSS triangle, because the known values are the three sides. The angles are
unknown. In this case, the formula for the area of the triangle is A =
c are the lengths of the three sides and =
s
s ( s − a )( s − b )( s − c ) , where a,b, and
1
( a + b + c ) . This is called Heron’s formula.
2
PRACTICE
4) Find the area of the triangle that has sides of length 5, 6, and 9