Use the Law of Cosines to solve triangles.

1
Objectives
Use the Law of Cosines to solve triangles. Apply
Area Formula to triangles.
2
Notes 1-4
Use the given measurements to solve ?ABC (nearest
tenth).
1. a 18, b 40, m C 82.5
2. x 18 y 10 z 9
3. Two model planes take off from the same spot.
The first plane travels 300 ft due west before
landing and the second plane travels 170 ft
southeast before landing. To the nearest foot,
how far apart are the planes when they land?
4. An artist needs to know the area of a
triangular piece of stained glass with sides
measuring 9 cm, 7 cm, and 5 cm. What is the area
to the nearest square centimeter?
3
In the previous lesson, you learned to solve
triangles by using the Law of Sines. However, the
Law of Sines cannot be used to solve triangles
for which side-angle-side (SAS) or side-side-side
(SSS) information is given. Instead, you must use
the Law of Cosines.
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Example 1 Using the Law of Cosines
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
a 8, b 5, m?C 32.2
Step 1 Find the length of the third side.
c2 a2 b2 2ab cos C
Law of Cosines
c2 82 52 2(8)(5) cos 32.2
Substitute.
c2 21.3
Use a calculator to simplify.
c 4.6
Solve for the positive value of c.
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Example 1 Continued
Step 2 Find the measure of the smaller angle, ?B.
Law of Sines
Substitute.
Step 3 Find the third angle measure.
m?A ? 112.4
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Example 2 Using the Law of Cosines
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
a 8, b 9, c 7
b2 a2 c2 2ac cos B
Law of cosines
92 82 72 2(8)(7) cos B
Substitute.
cos B 0.2857
Solve for cos B.
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Example 2 Continued
Use the given measurements to solve ?ABC (nearest
tenth).
Step 2 Find another angle measure
72 82 92 2(8)(9) cos C
Substitute Law of Cosines
cos C 0.6667
Solve for cos C.
Step 3 Find the third angle measure.
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When using the LAW of COSINES, find the largest
angle first. When using the LAW of SINES, find
the largest angle last (using the triangle sum
formula)
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Example 3
The surface of a hotel swimming pool is shaped
like a triangle with sides measuring 50 m, 28 m,
and 30 m. What is the area of the pools surface
to the nearest square meter?
Find the measure of the largest angle, ?A.
Law of Cosines
502 302 282 2(30)(28) cos A
?
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Notes
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
1. a 18, b 40, m C 82.5
2. x 18 y 10 z 9
11
Notes 3-4
3. Two model planes take off from the same spot.
The first plane travels 300 ft due west before
landing and the second plane travels 170 ft
southeast before landing. To the nearest foot,
how far apart are the planes when they land?
437 ft
4. An artist needs to know the area of a
triangular piece of stained glass with sides
measuring 9 cm, 7 cm, and 5 cm. What is the area
to the nearest square centimeter?
17 cm2