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

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

To find out how many solutions there are in an ambiguous case, compare the length of

#### 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

#### Example B

In

**Solution:** Again in this case,

There are two angles less than

To check to make sure

This problem yields two solutions. Either

#### Example C

A boat leaves 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

Now that we know the measure of

\begin{align*}\frac{\sin 65.1^\circ}{82} & = \frac{\sin 70.7^\circ}{a} \\ \frac{82 \sin 70.7^\circ}{\sin 65.1^\circ} & = a \\ a & = 85.3 \end{align*}

The boat is approximately 85.3 km away from lighthouse \begin{align*}B\end{align*}.

### Guided Practice

1. Prove using the Law of Sines: \begin{align*}\frac{a-c}{c} = \frac{\sin A - \sin C}{\sin C}\end{align*}

2. Find all possible measures of angle \begin{align*}B\end{align*} if any exist for the following triangle values: \begin{align*}A = 32.5^\circ, a = 26, b = 37\end{align*}

3. Find all possible measures of angle \begin{align*}B\end{align*} if any exist for the following triangle values: \begin{align*}A = 42.3^\circ, a = 16, b = 26\end{align*}

**Solutions:**

1. \begin{align*}\frac{\sin A}{a} & = \frac{\sin C}{c} \\ c \sin A & = a \sin C \\ c \sin A - c \sin C & = a \sin C - c \sin C \\ c (\sin A - \sin C) & = \sin C (a-c) \\ \frac{\sin A - \sin C}{\sin C} & = \frac{a-c}{c}\end{align*}

2. \begin{align*}\frac{\sin 32.5^\circ}{26} = \frac{\sin B}{37} \rightarrow B = 49.9^\circ\end{align*} or \begin{align*}180^\circ - 49.9^\circ = 130.1^\circ\end{align*}

3. no solution

### Concept Problem Solution

A drawing of this situation looks like this:

You can start by using the Law of Sines:

\begin{align*}\frac{\sin A}{a} = \frac{\sin B}{b}\end{align*}

and substitute known values:

\begin{align*}\frac{\sin 35}{14} = \frac{\sin B}{17}\end{align*}

Then solving for \begin{align*}\sin B\end{align*}:

\begin{align*}\sin B = \frac{17\sin 35^\circ}{14}\end{align*}

And so

\begin{align*}B \approx 44.15^\circ\end{align*}

Since the interior angles of any triangle add up to \begin{align*}180^\circ\end{align*}, we can find \begin{align*}\angle C \end{align*}:

\begin{align*}C = 180^\circ - 35^\circ - 44.15^\circ\\ C = 100.85\\ \end{align*}

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

\begin{align*} \frac{\sin A}{a} = \frac{\sin C}{c}\\ \frac{\sin 35}{14} = \frac{\sin 100.86}{c}\\ c = \frac{14 \sin 100.86}{\sin 35}\\ c = \frac{13.75}{.5735}\\ c = 23.976\\ \end{align*}

### Explore More

Find all possible measures of angle \begin{align*}B\end{align*} if any exist for each of the following triangle values.

- \begin{align*}A = 30^\circ, a = 13, b = 15\end{align*}
- \begin{align*}A = 42^\circ, a = 21, b = 12\end{align*}
- \begin{align*}A = 22^\circ, a = 36, b = 37\end{align*}
- \begin{align*}A = 87^\circ, a = 14, b = 12\end{align*}
- \begin{align*}A = 31^\circ, a = 25, b = 44\end{align*}
- \begin{align*}A = 59^\circ, a = 37, b = 41\end{align*}
- \begin{align*}A = 81^\circ, a = 22, b = 20\end{align*}
- \begin{align*}A = 95^\circ, a = 31, b = 34\end{align*}
- \begin{align*}A = 112^\circ, a = 12, b = 15\end{align*}
- \begin{align*}A = 78^\circ, a = 20, b = 16\end{align*}
- In \begin{align*}\triangle ABC\end{align*}, a=10 and \begin{align*}m\angle B=39^\circ\end{align*}. What's a possible value for b that would produce two triangles?
- In \begin{align*}\triangle ABC\end{align*}, a=15 and \begin{align*}m\angle B=67^\circ\end{align*}. What's a possible value for b that would produce no triangles?
- In \begin{align*}\triangle ABC\end{align*}, a=21 and \begin{align*}m\angle B=99^\circ\end{align*}. 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 \begin{align*}40^\circ\end{align*} east of north. Connie can see the school on a line \begin{align*}51^\circ\end{align*} 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 \begin{align*}30^\circ\end{align*} and the angle of elevation for Rose is \begin{align*}34^\circ\end{align*}. How high off the ground is the balloon?