H.Melikian/1200 1

7.2 Cosine Law and Area of Triangle

Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu

1. Determining if the Law of Sines or the Law of Cosines Should be Used to Begin to Solve an Oblique Triangle

2. Using the Law of Cosines to Solve the SAS Case3. Using the Law of Cosines to Solve the SSS Case4. Using the Law of Cosines to Solve Applied Problems

Involving Oblique Triangles5. Determining the Area of Oblique Triangles

6. Using Heron’s Formula to Determine the Area of an SSS Triangle

7. Solving Applied Problems Involving the Area of Triangles

H.Melikian/1200 2

The Law of Cosines

If A, B, and C are the measures of the angles of any triangle and if a, b, and c are the lengths of the sides opposite the corresponding angles, then

2 2 2

2 2 2

2 2 2

2 cos ,

2 cos ,

2 cos

a b c bc A

b a c ac B

c a b ab C

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Determining of the Law of Sines or Cosines Should Be Used to Begin to Solve an Oblique Triangle

Decide whether the Law of Sines or Cosines can be used to solve each triangle. Do not solve.a. b.

Law of Sines; given the lengths of two sides and an angle opposite one of the given sides.

Neither can be used.

83°

A

C

Bc = 15.2

b = 8.3

45°

68°

67°

H.Melikian/1200 4

Solving an SAS Oblique Triangle

Step 1: Use the Law of Cosines to determine the length of the missing side.

Step 2: Determine the measure of the smaller of the remaining two angles using the Law of Sines or using the alternate form of the Law of Cosines.

Step 3: Use the fact that the sum of the measures of the three angles of a triangle is 180 degrees to determine the measure of the remaining angle.

H.Melikian/1200 5

Solving a SAS TriangleSolve the given oblique triangle. Round all measurements to one decimal place.

Angles Sides

A = 61 a =

B = b = 20

C = c = 30

2 2 2

2 2 2

2 2

2 cos

20 30 2(20)(30)cos61

20 30 2(20)(30)cos61

26.8

a b c bc A

a

a

a

61°

b = 20

c = 30A

B

C

26.8

H.Melikian/1200 6

Solving a SAS Triangle-cont

Solve the given oblique triangle. Round all measurements to one decimal place.

Angles Sides

A = 61 a =

B = b = 20

C = c = 30

2 2 2

2 2 2

1

cos2

26.8 30 20

2(26.8)(30)

1218.24

16081218.24

cos 40.71608

a c bB

ac

B

61°

b = 20

c = 30A

B

C

26.8

180 61 40.7

78.3

C

C

H.Melikian/1200 7

Solving an SSS Oblique Triangle

Step 1: Use the alternate form of the Law of Cosines to determine the length of the largest angle. This is the angle opposite the longest side.

Step 2: Determine the measure of one of the remaining two angles using the Law of Sines or the alternate form of the Law of Cosines.

Step 3: Use the fact that the sum of the measures of the three angles of a triangle is 180 degrees to determine the measure of the remaining angle.

H.Melikian/1200 8

Solving a SSS Triangle

Solve oblique triangle ABC if a = 4, b = 3, and c = 6.

Angles Sides

A = a = 4

B = b = 3

C = c = 6

2 2 2

2 2 2

1

cos2

4 3 6

2(4)(3)

11

2411

cos 117.324

a b cC

ab

C

117.3

H.Melikian/1200 9

Solving a SSS Triangle-cont

Solve oblique triangle ABC if a = 4, b = 3, and c = 6.

Angles Sides

A = a = 4

B = b = 3

C = c = 6

sin sin

sin sin117.3

3 63sin117.3

sin6

26.4

B C

b cB

B

B

117.3

180 117.3 26.4 36.3C

153.6 degrees will not work.

H.Melikian/1200 10

Determining the Distance between Two AirplanesTwo planes take off from different runways at the same time. One plane flies at an average speed of 350 mph with a bearing of N 21 E. The other plane flies at an average speed of 420 mph with a bearing of S 84 W. How far apart are the planes from each other 2 hours after takeoff?

H.Melikian/1200 11

Determining the Distance between Two Airplanes

Two planes take off from different runways at the same time. One plane flies at an average speed of 350 mph with a bearing of N 21 E. The other plane flies at an average speed of 420 mph with a bearing of S 84 W. How far apart are the planes from each other 2 hours after takeoff?

a = 700, b = 840, and D = 117

The planes are about 1315 miles apart after 2 hours.

2 2 2

2 2 2

2 2

2 cos

700 840 2(700)(840)cos117

700 840 2(700)(840)cos117

1315.1

d a b ab D

d

a

a

H.Melikian/1200 12

Area of a Triangle

In any triangle, the area is given by

where b is the length of the base of the triangle and h is the length of the altitude drawn to that base (or drawn to an extension of that base).

1

2Area bh

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Area of a Triangle

If A, B, and C are the measures of the angles of any triangle and if a, b, and c are the lengths of the sides opposite the corresponding angles, then the area of triangle ABC is given by 1

sin21

sin21

sin2

Area bc A

Area ac B

Area ab C

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Determining the Area of an Oblique Triangle

Determine the area of the triangle.

34°

c = 22 ft

a = 20 ft

B

C

A

1sin

2Area ac B

1(20)(22)sin34

2Area

123.0 sq ft

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Determining the Area of an Oblique Triangle

Determine the area of the triangle.

1sin

2

(12)(17.6)sin135

74.7 sq cm

Area ac B

18°

135° a = 12 cm

B

C

A180 18 135

27

C

C

sin sin12

sin 27 sin1812sin 27

sin1817.6

c a

C Ac

c

c

H.Melikian/1200 16

Heron’s Formula

Suppose that a triangle has side lengths of a, b, and c. If the semiperimeter is then the area of the triangle is

1( )

2s a b c

( )( )( )Area s s a s b s c

H.Melikian/1200 17

Determining the Area of an SSS Oblique Triangle Using Heron’s Formula

Determine the area of the triangle. A

B

C

a = 28

c = 54b = 401( )

2s a b c

1(28 40 54)

261

( )( )( )Area s s a s b s c

61(61 28)(61 40)(61 54)

61(33)(21)(7) 543.98