Transcript 10.2 The Law of Sines

THE LAW OF SINES
2.3 I can solve triangles using the
Law of Sines
If none of the angles of a triangle is a right
angle, the triangle is called oblique.
All angles are acute
Two acute angles, one obtuse angle
To solve an oblique triangle means to
find the lengths of its sides and the
measurements of its angles.
FOUR CASES
CASE 1: One side and two angles are
known (SAA or ASA).
CASE 2: Two sides and the angle opposite
one of them are known (SSA).
Ambiguous Case
CASE 3: Two sides and the included
angle are known (SAS).
CASE 4: Three sides are known (SSS).
A
S
A
ASA
CASE 1: ASA or SAA
Use Law of Sines
S
A
A
SAA
S
A
S
CASE 2: SSA - Ambiguous Case
Use Law of Sines
S
A
S
CASE 3: SAS
Use Law of Cosine
S
S
S
CASE 4: SSS
Use Law of Cosines
Theorem Law of Sines
b  10sin   10sin 70  9.40
c  10sin   10sin 80  9.85
12 sin 20
a
 4.17
sin 100 
12 sin 60 
b
 10 .55
sin 100 
The Ambiguous Case: Case 2: SSA

The known information may result in
One triangle
Two triangles
No triangles
Not possible, so there is only one triangle!
sin 132 .5
a5
 7.37
sin 30
a  7.37, b  5, c  3,
  30,   17.5,   132.5
10 sin 45
sin  
 0.88
8
  62.1 or   117.9
1
2
Two triangles!!
Triangle 1:   62.1
  180  45  62.1  72.9
1
1
8 sin 72.9
a1 
 10.81
sin 45
  62.1,   72.9,   45
1
1
a1  10.81, b  8, c  10
Triangle 2:   117.9
  180  45  117.9  17.1
2
1
sin 17.1 sin 45

a2
8
8 sin 17.1
a2 
 3.33
sin 45
  117.9,   17.1,   45
2
2
a2  3.33, b  8, c  10
sin   1.28
No triangle with the given
measurements!
The Ambiguous Case: Case 2: SSA

The known information may result in
 One
triangle
 Two triangles
 No triangles
 The key to determining the possible triangles, if any,
lies primarily with the height, h and the fact h = b sin α
b
α
h
a
No Triangle

If a < h = b sin α, then side a is not sufficiently
long to form a triangle.
b
α

a h = b sinα
a < h = b sin α
One Right Triangle

If a = h = b sin α, then side a is just long enough to
form a triangle.
b
α

a h = b sinα
a = h = b sin α
Two Triangles

If a < b and h = b sin α < a, then two distinct
triangles can be formed
b
α

a
a
h = b sinα
a < b and h = b sin α < a
One Triangle

If a ≥ b, then only one triangle can be formed.
b
α


a
h = b sinα
a≥b
Fortunately we do not have to rely on the illustration to draw a
correct conclusion. The Law of Sines will help us.