7.6 Law of Sines. Use the Law of Sines to solve triangles and problems.
Law of sines
21
LAW OF SINES Sec. 5.5
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Sec. 5.5. Law of sines. Deriving the Law of Sines. C. In either triangle:. b. a. h. In the top triangle:. A. B. c. C. In the bottom triangle:. b. But, . h. a. so each of these last two e xpressions are equal!!!. A. c. B. Deriving the Law of Sines. C. b. a. - PowerPoint PPT Presentation
Transcript of Law of sines
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: sin hAb
In the top triangle: sin hBa
In the bottom triangle: sin hBa
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
sin hAb
sin hBa
Solve for h:
sinh b A sinh a B
Set equal:
sin sinb A a BWhich is equivalent to:
sin sinA Ba 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 Ca 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 Ba b
sin 36 sin 488 b
8sin 48sin36
b
10.115
Guided PracticeSolve , given the following.ABC
36A 48B 8a
A B
C
c
8b
36 48
sin sinA Ca c
sin 36 sin 968 c
8sin96sin36
c
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:
sin8215h
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 7328h
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 3117h
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 3821 25
C
C
211
125sin 38sin21
C
47.1
B38
2521
A
C
1 1180 94.9A B C 1
121sinsin 38
Aa
1
1
sinsin 3821
Aa
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
2221sinsin 38
Aa
2
2
sinsin 3821
Aa
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 10361A
16
10.772a 3.395b
Whiteboard PracticeSolve , given the following.ABC
49A 32a 28b
A B
C
c
322849
sin 49 sin32 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 5710 11
A
A
101
111sin 57sin10
A
67.3
B57
1110
C
A
1 1180 55.7C A B 1
110sinsin 57
Cc
1
1
sinsin 5710
Cc
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
2210sinsin 57
Cc
2
2
sinsin 5710
Cc
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
