ambiguous case aka – distinct triangles. we use this method to determine how many triangles can be...
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Ambiguous Case
AKA – Distinct Triangles
We use this method to determine how many triangles can be built with some given information.
How do we know when to use this method?
1. It will ask one of the following:
1. How many triangles…
2. How many distinct triangles…
3. How many different triangles…
The answer is always 0, 1, or 2. 0 is the least number that can be made and two is the most!
Steps:
1. Use law of sines to find unknown
2. Build table and fill in missing angles
3. Determine number of triangles from table
B
b
A
a
sinsin
Bsin
5
30sin
4
30sin5sin4 B
4
30sin5sin B
625.sin B
625.sinsinsin 11 B
396.38 B
1 2 3
Angle 1 is always the angle given!
30
30
Angle 2 is always the angle you found and its supplement!
39
180-39=141
141
Angle 3 is determined by the first two angles in each row. Are there enough degrees left to form a triangle?
30+39=69
180-69=111
11130+141=171
180-171=99
We were able to form 2 distinct triangles based on the chart!
PAGE 7
B
b
A
a
sinsin
Bsin
12
120sin
12
120sin12sin12 B
12
120sin12sin B
8660254038.sin B
8660.sinsinsin 11 B
60B
1 2 3
Angle 1 is always the angle given!
120
120
Angle 2 is always the angle you found and its supplement!
60
180-60=120
120
Angle 3 is determined by the first two angles in each row. Are there enough degrees left to form a triangle?
120+60=180
180-180=0
120+120=240
We were able to form 0 distinct triangles based on the chart!
PAGE 7
B
b
A
a
sinsin
Bsin
10
35sin
7
35sin10sin7 B
7
35sin10sin B
8193949091.sin B
8193.sinsinsin 11 B
5502.55 B
1 2 3
Angle 1 is always the angle given!
35
35
Angle 2 is always the angle you found and its supplement!
55
180-55=125
125
Angle 3 is determined by the first two angles in each row. Are there enough degrees left to form a triangle?
35+55=90
180-90=90
35+125=160
180-160=20
We were able to form 2 distinct triangles based on the chart!
90
20
PAGE 7
B
b
A
a
sinsin
Bsin
9
48sin
7
48sin9sin7 B
7
48sin9sin B
9554719185.sin B
9554.sinsinsin 11 B
738.72 B
1 2 3
Angle 1 is always the angle given!
48
48
Angle 2 is always the angle you found and its supplement!
73
180-39=141
107
Angle 3 is determined by the first two angles in each row. Are there enough degrees left to form a triangle?
48+73=121
180-121=59
5948+107=155
180-155=2525
We were able to form 2 distinct triangles based on the chart!
PAGE 7
B
b
A
a
sinsin
Bsin
36
45sin
40
45sin36sin40 B
40
45sin36sin B
6363961031.sin B
6364.sinsinsin 11 B
405.39 B
1 2 3
Angle 1 is always the angle given!
45
45
Angle 2 is always the angle you found and its supplement!
40
180-40=140
140
Angle 3 is determined by the first two angles in each row. Are there enough degrees left to form a triangle?
45+40=85
180-85=95
95
45+140=185
We were able to form 2 distinct triangles based on the chart!
PAGE 7
C
c
A
a
sinsin
Csin
18
30sin
5
30sin18sin5 C
5
30sin18sin C
8.1sin C
8.1sinsinsin 11 C
ERRORC
We were able to form 0 distinct triangles.
PAGE 8
A
a
V
v
L
l
sinsinsin
A
a
V
v
sinsin
5.44sin
3.11
sin
12.13
V
5.44sin12.13sin3.11 V
3.11
5.44sin12.13sin V
813799075.sin V
813799075.sinsinsin 11 V
46.54A
4.54A
9.985.444.54
1.819.98180
PAGE 6
Homework
•Page 8
#1-5,7