Additional Topics in Trigonometryteachers.dadeschools.net/rcarrasco/bzpc5e_06_01.pdf · Oblique...

Chapter 6

Additional Topics

in Trigonometry

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

6.1 The Law of Sines


Objectives:

• Use the Law of Sines to solve oblique triangles.

• Use the Law of Sines to solve, if possible, the

triangle or triangles in the ambiguous case.

• Find the area of an oblique triangle using the sine

function.

• Solve applied problems using the Law of Sines.


Oblique Triangles

An oblique triangle is a triangle that does not contain

a right angle.

An oblique triangle has

either three acute angles

or two acute angles and

one obtuse angle.

The relationships among the sides

and angles of right triangles

defined by the trigonometric

functions are not valid for

oblique triangles.


The Law of Sines


Solving Oblique Triangles

Solving an oblique triangle means finding the lengths

of its sides and the measurements of its angles.

The Law of Sines can be used to solve a triangle in

which one side and two angles are known. The three

known measurements can be abbreviated using SAA (a

side and two angles are known) or ASA (two angles and

the side between them are known).


Example: Solving an SAA Triangle Using the Law of Sines

Solve the triangle with A = 64°, C = 82°, and c = 14

centimeters. Round lengths of sides to the nearest tenth.

180A B C

64 82 180B

146 180B

34B

sin sin

a c

A C

sin sina C c A

sin

sin

c Aa

C

14sin64

sin82

12.7 cm


Example: Solving an SAA Triangle Using the Law of Sines (continued)

Solve the triangle with A = 64°, C = 82°, and c = 14

centimeters. Round lengths of sides to the nearest tenth.

sin sin

b c

B C

sin sinb C c B

sin

sin

c Bb

C

14sin34

sin82

7.4 cm

34

12.7 cm

7.4 cm

B

a

b


Example: Solving an ASA Triangle Using the Law of Sines

Solve triangle ABC if A = 40°, C = 22.5°, and b = 12.

Round measures to the nearest tenth.

A

B

C12

180A B C

40 22.5 180B

62.5 180B

117.5B

sin sin

b a

B A

sin sinb A a B

sin

sin

b Aa

B

12sin 40

sin117.5

8.7


Example: Solving an ASA Triangle Using the Law of Sines (continued)

Solve triangle ABC if A = 40°, C = 22.5°, and b = 12.

Round measures to the nearest tenth.

A

B

C12

sin sin

b c

B C

sin sinc B b C

sin

sin

b Cc

B

12sin 22.5

sin117.5

5.2

117.5

8.7

5.2

B

a

c


The Ambiguous Case (SSA)

If we are given two sides and an angle opposite one of

the two sides (SSA), the given information may result in

one triangle, two triangles, or no triangle at all.

SSA is known as the ambiguous case when using the

Law of Sines because the given information may result

in one triangle, two triangles, or no triangle at all.


The Ambiguous Case (SSA) (continued)


Example: Solving an SSA Triangle Using the Law of Sines (No Solution)

Solve triangle ABC if A = 50°, a = 10, and b = 20.

sin sin

a b

A B

sin sina B b A

sinsin

b AB

a

20sin50

10

1.53

There is no angle B for which the sine is greater than 1.

There is no triangle with the given measurements.


Example: Solving an SSA Triangle Using the Law of Sines (Two Solutions)

Solve triangle ABC if A = 35°, a = 12, and b = 16. Round

lengths of sides to the nearest tenth and angle measures to

the nearest degree.

sin sin

a b

A B

sin sina B b A

sinsin

b AB

a

16sin35

12

0.7648

1sin 0.7648 50 There are two angles between 0° and

180° for which sinB = 0.7648


Example: Solving an SSA Triangle Using the Law of Sines (Two Solutions) (continued)



the nearest degree.

1sin 0.7648 50

1 50B

2 180 50 130B

1 35 50 85A B

2 35 130 165A B

there are two possible solutions.

1 2180 and 180 ,A B A B





the nearest degree.

1 50B 2 130B

1 1 180A B C 2 2 180A B C

135 50 180C

185 180C

1 95C

235 130 180C

2165 180C

2 15C





the nearest degree.

1 50B 1 95C 2 130B 2 15C

1

1 1sin sin

b c

B C

2

2 2sin sin

b c

B C

1 1 1sin sinb C c B2 2 2sin sinb C c B

11

1

sin

sin

b Cc

B

22

2

sin

sin

b Cc

B

16sin95

sin50

20.8

16sin15

sin130

5.4





the nearest degree.

There are two triangles. In one triangle, the solution is

In the other triangle, the solution is

1 50B 1 95C 1 20.8c

2 130B 2 15C 2 5.4c


The Area of an Oblique Triangle

The area of a triangle equals one-half the product of the

lengths of two sides times the sine of their included angle.

In the figure, this wording can be expressed by the

formulas

1 1 1Area sin sin sin

2 2 2bc A ab C ac B


Example: Finding the Area of an Oblique Triangle

Find the area of a triangle having two sides of length 8

meters and 12 meters and an included angle of 135°.

Round to the nearest square meter.

8 m 12 m135

A B

C1

Area sin2

ab C

1(12)(8)(sin135 )

2

34 sq m


Example: Application

Two fire-lookout stations are 13 miles apart, with station B

directly east of station A. Both stations spot a fire. The

bearing of the fire from station A is N35°E and the bearing

of the fire from station B is N49°W. How far, to the

nearest tenth of a mile, is the fire from station B?

A B

C

3549

13

90 35 55A

90 49 41B

180A B C

55 41 180C

96 180C 84C


Example: Application (continued)

Two fire-lookout stations are 13 miles apart, with station B directly

east of station A. Both stations spot a fire. The bearing of the fire

from station A is N35°E and the bearing of the fire from station B is

N49°W. How far, to the nearest tenth of a mile, is the fire from

station B?

A B

C

3549

sin sin

c a

C A

sin sinc A a C

sin

sin

c Aa

C

13sin55

sin84

11

The fire is approximately 11 miles from station B.

Additional Topics in Trigonometryteachers.dadeschools.net/rcarrasco/bzpc5e_06_01.pdf · Oblique...

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