1. 9.3 The Law of Sines AMBIGUOUS CASE

2. Activity 1 For this activity, use a ruler, compass, and protractor.
Draw A with measure 30o.
Along one ray of A, locate point C 10 cm from point A.
For each of the following compass settings, draw a large arc.
Then tell whether the arc crosses the other ray of A and, if so, in how many points.
Compass at C and opened to 4 cm
Compass at C and opened to 5 cm
Compass at C and opened to 6 cm
Activity 1 shows that when you are given the lengths of two sides of a triangle and the measure of a nonincluded angle (SSA), it may be possible to construct no triangle, one triangle, or two triangles. For this reason the SSA situation is called the ambiguous case. no
yes, 1
yes, 2

3. Activity 2
Show that your answers to Activity 1 agree with what the law of sines would give in each of the following SSA situations.
If A = 30o, b = 10, and a = 4, find B.
If A = 30o, b = 10, and a = 5, find B.
If A = 30o, b = 10, and a = 6, find B.
No solution

4. Ambiguous Case for Acute Angle A

5. Ambiguous Case for Obtuse Angle A

6. Ambiguous Case
If given the lengths of two sides and the angle opposite one of them, it is possible that 0, 1, or 2 such triangles exist.
Some basic facts that should be kept in mind:
For any angle , –1  sin   1, if sin  = 1, then  = 90o and the triangle is a right triangle.
2. sin  = sin(180o –  ).
3. The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-value angle is opposite the intermediate side (assuming unequal sides).

7. Possible Outcomes Outcome 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
In this case, when applying the Law of Sines, you may get sinB > 1. A B c
NO SOLUTION

8. Possible Outcomes Outcome 1: If A is acute and a < b
b) If a = b sinA C
h = b sin A b h
= a A c B
1 SOLUTION

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

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

11. Possible Outcomes Outcome 2: If A is obtuse and a ≤ b
In this case, when applying the Law of Sines, you may get sinB > 1.
NO SOLUTION

12. Solving the Ambiguous Case: No Such Triangle
Case 2 Two sides and one angle not included between the sides known:
SSA
Example 3: Solve the triangle ABC if B = 55°40´,
b = 8.94 meters, and a = 25.1 meters.
[Solution] Use the law of sines to find A.
Since sin A cannot be greater than 1, the triangle does
not exist.

13. Solving the Ambiguous Case: Two Triangles
Case 2 Two sides and one angle not included between the sides known:
SSA
Example 4:
Solve the triangle ABC if A = 55.3o,
a = 22.8 feet, and b = 24.9 feet.
[Solution]

14. Solving the Ambiguous Case: No Such Triangle
To see if B2 = 116.1o is a valid possibility, add 116.1o
to the measure of A: 116.1o o = 171.4o. Since
this sum is less than 180o, it is a valid triangle.
Now separate the triangles into two: AB1C1 and AB2C2.
60.8o

15. Solving the Ambiguous Case: No Such Triangle
Now solve for triangle AB2C2.

16. Number of Triangles Satisfying the Ambiguous Case
Let sides a and b and angle A be given in triangle ABC. (The
law of sines can be used to calculate sin B.)
If sin B > 1, then no triangle satisfies the given conditions.
If sin B = 1, then one triangle satisfies the given conditions and B = 90°.
If 0 < sin B < 1, then either one or two triangles satisfy the given conditions
If sin B = k, then let B1 = sin-1 k and use B1 for B in the first triangle.
Let B2 = 180° – B1. If A + B2 < 180°, then a second triangle exists. In this case, use B2 for B in the second triangle.

17. Assignment
P. 348 #7, 13, 17, 21, 22