The Law of Cosines

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13-6
The Law of Cosines
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Find each measure to the nearest
tenth. 1. m?y 2. x 3. y 4. What is the
area of ?XYZ ? Round to the nearest square
unit.
8.8
104
18.3
60 square units
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Objectives
Use the Law of Cosines to find the side lengths
and angle measures of a triangle. Use Herons
Formula to find the area of a triangle.
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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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Write an equation that relates the side lengths
of ?DBC.
a2 (b x)2 h2
Pythagorean Theorem
Expand (b x)2.
a2 b2 2bx x2 h2
In ?ABD, c2 x2 h2. Substitute c2 for x2 h2.
a2 b2 2bx c2
a2 b2 2b(c cos A) c2
a2 b2 c2 2bccos A
The previous equation is one of the formulas for
the Law of Cosines.
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Example 1A 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 1A Continued
Step 2 Find the measure of the smaller angle, ?B.
Law of Sines
Substitute.
Solve for sin B.
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Example 1A Continued
Step 3 Find the third angle measure.
m?A 35.4 32.2 ? 180
Triangle Sum Theorem
m?A ? 112.4
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Example 1B 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 1B Continued
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
Step 2 Find another angle measure
c2 a2 b2 2ab cos C
Law of cosines
72 82 92 2(8)(9) cos C
Substitute.
cos C 0.6667
Solve for cos C.
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Example 1B Continued
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
Step 3 Find the third angle measure.
Triangle Sum Theorem
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Check It Out! Example 1a
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
Step 1 Find the length of the third side.
a2 b2 c2 2bc cos A
Law of Cosines
a2 232 182 2(23)(18) cos 173
Substitute.
a2 1672.8
Use a calculator to simplify.
a 40.9
Solve for the positive value of c.
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Check It Out! Example 1a Continued
Step 2 Find the measure of the smaller angle, ?C.
Law of Sines
Substitute.
Solve for sin C.
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Check It Out! Example 1a Continued
Step 3 Find the third angle measure.
Triangle Sum Theorem
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Check It Out! Example 1b
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
a 35, b 42, c 50.3
c2 a2 b2 2ab cos C
Law of cosines
50.32 352 422 2(35)(50.3) cos C
Substitute.
cos C 0.1560
Solve for cos C.
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Check It Out! Example 1b Continued
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
a 35, b 42, c 50.3
Step 2 Find another angle measure
a2 c2 b2 2cb cos A
Law of cosines
352 50.32 422 2(50.3)(42) cos A
Substitute.
cos A 0.7264
Solve for cos A.
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Check It Out! Example 1b Continued
Step 3 Find the third angle measure.
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Example 2 Problem-Solving Application
If a hiker travels at an average speed of 2.5
mi/h, how long will it take him to travel from
the cave to the waterfall? Round to the nearest
tenth of an hour.
The answer will be the number of hours that the
hiker takes to travel to the waterfall.
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The answer will be the number of hours that the
hiker takes to travel to the waterfall.
List the important information

• The cave is 3 mi from the cabin.

• The waterfall is 4 mi from the cabin. The path
  from the cabin to the waterfall makes a 71.7
  angle with the path from the cabin to the cave.

• The hiker travels at an average speed of 2.5 mi/h.

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Use the Law of Cosines to find the distance d
between the water-fall and the cave. Then
determine how long it will take the hiker to
travel this distance.
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Step 1 Find the distance d between the waterfall
and the cave.
d2 c2 w2 2cw cos D
Law of Cosines
Substitute 4 for c, 3 for w, and 71.7 for D.
d2 42 32 2(4)(3)cos 71.7
d2 17.5
Use a calculator to simplify.
d 4.2
Solve for the positive value of d.
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Step 2 Determine the number of hours.
In 1.7 h, the hiker would travel 1.7 ? 2.5 4.25
mi. Using the Law of Cosines to solve for the
angle gives 73.1. Since this is close to the
actual value, an answer of 1.7 hours seems
reasonable.
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Check It Out! Example 2
A pilot is flying from Houston to Oklahoma City.
To avoid a thunderstorm, the pilot flies 28 off
the direct route for a distance of 175 miles. He
then makes a turn and flies straight on to
Oklahoma City. To the nearest mile, how much
farther than the direct route was the route taken
by the pilot?
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The answer will be the additional distance the
pilot had to fly to reach Oklahoma City.
List the important information

• The direct route is 396 miles.

• The pilot flew for 175 miles off the course at an
  angle of 28 before turning towards Oklahoma City.

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Use the Law of Cosines to find the distance from
the turning point on to Oklahoma City. Then
determine the difference additional distance and
the direct route.
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Step 1 Find the distance between the turning
point and Oklahoma City. Use side-angle-side.
b2 c2 a2 2ca cos B
Law of Cosines
b2 3962 1752 2(396)(175)cos 28
Substitute 396 for c, 175 for a, and 28 for B.
b2 65072
Use a calculator to simplify.
b 255
Solve for the positive value of b.
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Step 2 Determine the number of additional miles
the plane will fly.
Add the actual miles flown and subtract from that
normal distance to find the extra miles flown
Total miles traveled.
255 175 430
430 396 34
Additional miles.
By using the Law of Cosines the length of the
extra leg of the trip could be determined.
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The Law of Cosines can be used to derive a
formula for the area of a triangle based on its
side lengths. This formula is called Herons
Formula.
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Example 3 Landscaping Application
A garden has a triangular flower bed with sides
measuring 2 yd, 6 yd, and 7 yd. What is the area
of the flower bed to the nearest tenth of a
square yard?
Step 1 Find the value of s.
Use the formula for half of the perimeter.
Substitute 2 for a, 6 for b, and 7 for c.
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Example 3 Continued
Step 2 Find the area of the triangle.
Herons formula
Substitute 7.5 for s.
A 5.6
Use a calculator to simplify.
The area of the flower bed is 5.6 yd2.
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Example 3 Continued
Check Find the measure of the largest angle, ?C.
c2 a2 b2 2ab cos C
Law of Cosines
72 22 62 2(2)(6) cos C
Substitute.
Solve for cos C.
cos C 0.375
?
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Check It Out! 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?
Step 1 Find the value of s.
Use the formula for half of the perimeter.
Substitute 50 for a, 28 for b, and 30 for c.
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Check It Out! Example 3 Continued
Step 2 Find the area of the triangle.
Herons formula
Substitute 54 for s.
Use a calculator to simplify.
A 367 m2
The area of the flower bed is 367 m2.
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Check It Out! Example 3 Continued
Check Find the measure of the largest angle, ?A.
502 c2 b2 2cb cos A
Law of Cosines
502 302 282 2(30)(28) cos A
Substitute.
Solve for cos A.
cos A 0.4857
?
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Lesson Quiz Part I
Use the given measurements to solve ?ABC. Round
to the nearest tenth.
1. a 18, b 40, m C 82.5
2. a 18.0 b 10 c 9
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Lesson Quiz Part II
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