7.2 The Ambiguous Case of the Law of Sines

Direct Application of Law of Sines applies for SAA or ASA

If given Case 2, SSA, then zero, one or two such triangles may exist, because SSA does not prove congruence between two triangles.

If angle A is acute, there are four possibilities

Number of Triangles	Sketch	Applying Law of Sines Leads to
0
1
1
2

If angle A is obtuse, there are two possible outcomes

Number of Triangles	Sketch	Applying Law of Sines Leads to
0
1

Applying the Law of Sines

1. For any angle of a triangle, . If , then and the triangle is a right triangle.

2. (Supplementary angles have the same sine value.)

3. The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-valued angle is opposite the intermediate side. (assuming the triangle has sides that are all of different lengths.)

Example

Solve triangle ABC if

Example

Solve triangle ABC if

Number of Triangles Satisfying the Ambiguous Case (SSA)

Let sides a and b and angle A be given in triangle ABC.

(The law of sines can be used to calculate the value of sin B.)

1. If applying the law of sines results in an equation having , then

no triangle satisfies the given conditions.

2. If , then one triangle satisfies the given conditions and .

3. If , then either one or two triangles satisfy the given conditions.

a) If

b) Let then a second triangle exists.

In this case, use in the second triangle.

Outline the steps to take to solve each triangle. Do not actually solve the triangle.

1.

2.

3.

4.

5.

Example

Solve triangle ABC, given

Example

Without using the law of sines, explain why no triangle ABC exists satisfying