While working in art class you are trying to design pieces of glass that you will eventually fit together into a sculpture. You are drawing out what you think will be a diagram of one of the pieces. You have a side of length 14 inches, and side of length 17 inches, and an angle next to the 17 in side of (not the angle between the 14 in and 17 in pieces, but the one at the other end of the 17 in piece). Your diagram looks like this:

It occurs to you that you could use your knowledge of math to find out if you are going to be able to finish the drawing and make a piece that could actually be built.

Can you figure out how to do this?

By the end of this Concept, you'll know how to apply the Law of Sines to determine the number of possible solutions for a triangle.

### Watch This

James Sousa: The Law of Sines: The Ambiguous Case

### Guidance

In below, we know two sides and a non-included angle. Remember that the Law of Sines states: . Since we know , and , we can use the Law of Sines to find . However, since this is the SSA case, we have to watch out for the Ambiguous case. Since , we could be faced with situations where either no possible triangles exist, one possible triangle exists, or two possible triangles exist.

To find out how many solutions there are in an ambiguous case, compare the length of to . If , then there are no solutions. If , then there is one solution. If , then there are two solutions.

#### Example A

Find .

**
Solution:
**
Use the Law of Sines to determine the angle.

**
Since no angle exists with a sine greater than 1, there is no solution to this problem.
**

We also could have compared and beforehand to see how many solutions there were to this triangle.

: since which tells us there are no solutions.

#### Example B

In , and . Find .

**
Solution:
**
Again in this case,
and we know two sides and a non-included angle. By comparing
and
, we find that
. Since
we know that there will be two solutions to this problem.

There are two angles less than with a sine of 0.6666667, however. We found the first one, , by using the inverse sine function. To find the second one, we will subtract from .

To check to make sure is a solution, we will use the Triangle Sum Theorem to find the third angle. Remember that all three angles must add up to .

This problem yields two solutions. Either or .

#### Example C

A boat leaves lighthouse and travels 63km. It is spotted from lighthouse , which is 82km away from lighthouse . The boat forms an angle of with both lighthouses. How far is the boat from lighthouse ?

**
Solution:
**
In this problem, we again have the SSA angle case. In order to find the distance from the boat to the lighthouse (a) we will first need to find the measure of
. In order to find
, we must first use the Law of Sines to find
. Since
, this situation will yield exactly one answer for the measure of
.

Now that we know the measure of , we can find the measure of angle . Finally, we can use to find side .

The boat is approximately 85.3 km away from lighthouse .

### Vocabulary

**
Law of Cosines:
**
The
**
law of cosines
**
is an equation relating the length of one side of a triangle to the lengths of the other two sides and the sine of the angle included between the other two sides.

**
Law of Sines:
**
The
**
law of sines
**
is a rule applied to triangles stating that the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle in the triangle to the side opposite that angle.

### Guided Practice

1. Prove using the Law of Sines:

2. Find all possible measures of angle if any exist for the following triangle values:

3. Find all possible measures of angle if any exist for the following triangle values:

**
Solutions:
**

1.

2. or

3. no solution

### Concept Problem Solution

A drawing of this situation looks like this:

You can start by using the Law of Sines:

and substitute known values:

Then solving for :

And so

Since the interior angles of any triangle add up to , we can find :

This information can be used again in the Law of Sines:

### Practice

Find all possible measures of angle if any exist for each of the following triangle values.

- In , a=10 and . What's a possible value for b that would produce two triangles?
- In , a=15 and . What's a possible value for b that would produce no triangles?
- In , a=21 and . What's a possible value for b that would produce one triangle?
- Bill and Connie are each leaving for school. Connie's house is 4 miles due east of Bill's house. Bill can see the school in the direction east of north. Connie can see the school on a line west of north. What is the straight line distance of each person from the school?
- Rochelle and Rose are each looking at a hot air balloon. They are standing 2 miles apart. The angle of elevation for Rochelle is and the angle of elevation for Rose is . How high off the ground is the balloon?