1. The Ambiguous Case for the Law of Sines
5.7
The Ambiguous Case for the Law of Sines

2. AMBIGUOUS Open to various interpretations Having double meaning
Difficult to classify, distinguish, or comprehend
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3. Opposite sides of angles of a triangle
RECALL:
Opposite sides of angles of a triangle
Interior Angles of a Triangle Theorem
Triangle Inequality Theorem
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4. Triangles that do not have right angles
RECALL:
Oblique Triangles
Triangles that do not have right angles
(acute or obtuse triangles)
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5. RECALL:
LAW OF SINE
– 1  sin   1
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6. Sine values of supplementary angles are equal.
RECALL:
Sine values of supplementary angles are equal.
Example:
Sin 80o =
Sin 100o =
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7. Law of Sines: The Ambiguous Case
Given:
lengths of two sides and the angle opposite one of them (S-S-A)
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8. Possible Outcomes Case 1: If A is acute and a < b
a. If a < b sinA a C b a
h = b sin A b B A h c A B c
NO SOLUTION

9. Possible Outcomes Case 1: If A is acute and a < b b. If a = b sinA
h = b sin A b h
= a A c B
1 SOLUTION

10. Possible Outcomes Case 1: If A is acute and a < b
h = b sin A
b. If a > b sinA C b a h a  
180 -  A B B c
2 SOLUTIONS

11. Possible Outcomes Case 2: If A is obtuse and a > b C a b A c B
ONE SOLUTION

12. Possible Outcomes Case 2: If A is obtuse and a ≤ b C a b A c B
NO SOLUTION

13. Determine the number of possible solutions for each triangle.
i) A=30deg a=8 b=10
ii) b=8 c = 10 B = 118 deg

14. Find all solutions for each triangle.
i) a = 4 b = 3 A = 112 degrees
ii) A = 51 degrees a = 40 c = 50

15. EXAMPLE 1 Given: ABC where a = 22 inches b = 12 inches a>b
mA = 42o
a>b
mA > mB
SINGLE–SOLUTION CASE
(acute)
Find m B, m C, and c.

16. sin A = sin B a b Sin B  0.36498 mB = 21.41o or 21o
Sine values of supplementary angles are equal.
The supplement of B is B2.  mB2=159o

17. mC = 180o – (42o + 21o) mC = 117o
sin A = sin C a c
c = inches
SINGLE–SOLUTION CASE

18. sin A = sin B a b Sin B  1.66032 mB = ? Sin B > 1 NOT POSSIBLE !
Recall: – 1  sin   1
NO SOLUTION CASE
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19. EXAMPLE 3 Given: ABC where b = 15.2 inches a = 20 inches b < a
mB = 110o
b < a
NO SOLUTION CASE
(obtuse)
Find m B, m C, and c.

20. sin A = sin B a b Sin B  1.23644 mB = ? Sin B > 1 NOT POSSIBLE !
Recall: – 1  sin   1
NO SOLUTION CASE
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21. EXAMPLE 4 Given: ABC where a = 24 inches b = 36 inches a < b
mA = 25o
a < b
a ? b sin A
24 > 36 sin 25o
TWO – SOLUTION CASE
(acute)
Find m B, m C, and c.

22. sin A = sin B a b Sin B  0.63393 mB = 39.34o or 39o
The supplement of B is B2.  mB2 = 141o
mC1 = 180o – (25o + 39o) mC1 = 116o
mC2 = 180o – (25o+141o) mC2 = 14o

23. sin A = sin C a c1 c1 = 51.04 inches sin A = sin C a c2
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24. EXAMPLE 3 Final Answers: mB1 = 39o mC1 = 116o c1 = 51.04 in.
TWO – SOLUTION CASE
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25. Find m B, m C, and c, if they exist.
SEATWORK: (notebook)
Answer in pairs.
Find m B, m C, and c, if they exist.
 1) a = 9.1, b = 12, mA = 35o
 2) a = 25, b = 46, mA = 37o
3) a = 15, b = 10, mA = 66o  
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26. Answers: 1)Case 1: mB=49o,mC=96o,c=15.78 Case 2:
2)No possible solution.
3)mB=38o,mC=76o,c=  
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