LAW OF SINES

Sec. 5.5

Deriving the Law of Sines

A

A

B

B

C

Cc

c

a

ab

b

h

h

In either triangle: sinh

Ab

In the top triangle: sinh

Ba

In the bottom triangle: sinh

Ba

But, sin sinB B so each of these last twoexpressions are equal!!!

Deriving the Law of Sines

A

A

B

B

C

Cc

c

a

ab

b

h

h

sinh

Ab

sinh

Ba

Solve for h:

sinh b A sinh a B

Set equal:

sin sinb A a BWhich is equivalent to:

sin sinA B

a b

Law of SinesIn any triangle with angles A, B, and Copposite sides a, b, and c, respectively, the followingequation is true:

ABC

sin sin sinA B C

a b c

The Law of Sines works most easily withthese two triangle cases: AAS, ASA

Guided PracticeSolve , given the following.ABC

36A 48B 8a

A B

C

c

8b

36 48

180 36 48C 96 sin sinA B

a b

sin 36 sin 48

8 b

8sin 48

sin36b

10.115

Guided PracticeSolve , given the following.ABC

36A 48B 8a

A B

C

c

8b

36 48

sin sinA C

a c

sin 36 sin 96

8 c

8sin96

sin36c

13.536

The Ambiguous Case (SSA)We wish to construct ABC given angle A, side AB, & side BC.

1. Suppose angle A is obtuse and that side AB is as shownbelow. To complete the triangle, side BC must determine apoint on the dotted horizontal line (which extends infinitely tothe left). Explain from the picture why a unique triangle ABCis determined if BC > AB, but no triangle is determined ifBC < AB.

A

B

The Ambiguous Case (SSA)We wish to construct ABC given angle A, side AB, & side BC.

2. Suppose angle A is acute and that side AB is as shownbelow. To complete the triangle, side BC must determine apoint on the dotted horizontal line (which extends infinitely tothe right). Explain from the picture why a unique triangle ABCis determined if BC = h, but no triangle is determined if BC < h.

A

B

h

The Ambiguous Case (SSA)We wish to construct ABC given angle A, side AB, & side BC.

3. Suppose angle A is acute and that side AB is as shownbelow. If AB > BC > h, then we can form a triangle as shown.Find a second point C on the dotted horizontal line that givesa side BC of the same length, but determines a differenttriangle. (This is the “ambiguous case.”)

A

B

h

C C

The Ambiguous Case (SSA)We wish to construct ABC given angle A, side AB, & side BC.

4. Explain why sin(C) is the same in both triangles in theambiguous case. (This is why the Law of Sines is alsoambiguous in this case.)

A

B

h

C C

5. Explain from the figure below why a unique triangle isdetermined if BC > AB.

Guided PracticeState whether the given measurements determine zero, one, ortwo triangles.

B

(a) B = 82 , b = 17, c = 15

82

15 17

A

C

h

Solve for h:

sin8215

h

15sin82h 14.9

Because h < c < b, one triangle is formed.

Practice ProblemsState whether the given measurements determine zero, one, ortwo triangles.

A

(b) A = 73 , a = 24, b = 28

73

28 24

C

B

h

Solve for h:

sin 7328

h

28sin 73h 26.8

Because a < h, no triangle is formed.

Practice ProblemsState whether the given measurements determine zero, one, ortwo triangles.

C

(c) C = 31 , a = 17, c = 10

31

17 10

B

A

h

Solve for h:

sin 3117

h

17sin31h 8.756

Because h < c < a, two triangles are formed.

Guided Practice Two triangles can be formed using the given measurements.Solve both triangles.

B

(a) B = 38 , b = 21, c = 25

38

2521

A

C

Here, C is acute:

1sinsin 38

21 25

C

C

211

1

25sin 38sin

21C

47.1

B38

2521

A

C

1 1180 94.9A B C 1

1

21sin

sin 38

Aa

1

1

sinsin 38

21

A

a

33.987

Guided Practice Two triangles can be formed using the given measurements.Solve both triangles.

B

(a) B = 38 , b = 21, c = 25

38

2521

A

C

What if C is obtuse?

C

21

2 1180 132.9C C

22

21sin

sin 38

Aa

2

2

sinsin 38

21

A

a

5.414

B38

25

A

C

21

2 2180 9.1A B C

Whiteboard PracticeSolve , given the following.ABC

50A 62B 4a

A B

C

c

4b

50

68C

62

4.610b4.841c

Whiteboard PracticeSolve , given the following.ABC

16B 103C 12c

A B

C

12

ab 103

61A

16

10.772a 3.395b

Whiteboard PracticeSolve , given the following.ABC

49A 32a 28b

A B

C

c

3228

49

sin 49 sin

32 28

B

1 28sin 49sin32

B

41.3

180 49 41.3C 89.7 sin89.7 sin 49

32c

32sin89.7

sin 49c

42.400

Whiteboard PracticeSolve , given the following.ABC

103C 46b 61c

A

B

C

61

a

46103

47.3B 29.7A 31.029a

Whiteboard ProblemsTwo triangles can be formed using the given measurements.Solve both triangles.

B

(b) B = 57 , a = 11, b = 10

57

1110

C

A

Here, A is acute:

1sinsin 57

10 11

A

A

101

1

11sin 57sin

10A

67.3

B57

1110

C

A

1 1180 55.7C A B 1

1

10sin

sin 57

Cc

1

1

sinsin 57

10

C

c

9.850

Whiteboard ProblemsTwo triangles can be formed using the given measurements.Solve both triangles.

What if A is obtuse?

2 1180 112.7A A

22

10sin

sin 57

Cc

2

2

sinsin 57

10

C

c

2.132

B57

11

C

A

10

2 2180 10.3C B A

B57

1110

C

AA

10

(b) B = 57 , a = 11, b = 10