Have you ever watched a movie with your friends and realized that each of you interpreted some scenes quite differently? This kind of thing happens all the time, right? If you and your friends disagreed about many different aspects of the film, you might conclude that the filmmakers decided to be ambiguous about the movie's meaning or message, leaving interpretation of events, characters, and symbols to viewers.
The two types of problems on the previous page definitely relate to The Law of Sines. There are several ambiguous cases, though, in which the Law of Sines may (or may not) apply. Watch the video below to learn how to "interpret" and solve these types of problems.
When you know the length of two sides of a triangle and the measure of one of its angles, there are three possible things which can happen:
- You create 1 triangle.
- You create 2 triangles.
- No triangle exists.
The fact that there are three possible outcomes makes the situation or problem ambiguous.
Watch the following video to learn more.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
When using the Law of Sines to solve a triangle with two sides and an excluded angle given, we must be careful looking for missing sides due to ambiguity. That means, the given information does not always produce a clear result in this case. We can actually encounter three scenarios based on the information. Let’s look at this dynamic diagram for a moment. Notice we have two side lengths and an excluded angle measure given. With this information, we have some room to play with the pieces to try to form a triangle. A hinge, more-or-less is created at the point C, and we can swing the side b to try to form a triangle without changing the side lengths or given angle measure.
However, if side length b is too short, say four units in length, we cannot connect it to side length c. Therefore a triangle cannot be created. If we extend b to be five point 4 units long, the side swings down to connect with the side c, and forms a right triangle. But only ONE triangle is formed. If we make b just a little longer, we can see that two triangles are formed. But if b gets too large, then only one triangle can be formed again since the angle measure must remain constant and in the triangle on the left, it is the supplement of angle beta used in the construction of the triangle.
To formally identify which scenario you encounter, we can consider a formula to help you determine whether side b is too long, too short, or just right to form a triangle with the given information… The relationship between the length of b and the height of the triangle is an important one. Fortunately, we can quickly find the height of a given triangle using the formula, h = a * sin(β).
No triangle is created when b > h. One triangle is created when b = h or b > a, and two triangles are created when a > b > h. You’ll remember that when b = 4, it was too short to create a triangle. Let’s check our formula to see whether that supports our findings. H is equal to seven times the sine of fifty degrees, which is approximately five point four. Since b is less than h, we cannot create a triangle. In the second example h is equal to seven times the sine of fifty degrees, or again five point four. In this case h is equal to b and therefore one triangle is created. Finally, in scenario three, we see h is equal to seven times the sine of fifty degrees, or again five point four. Since b is longer than h and shorter than a, then two triangles can be created. Do you remember what happens if b is larger than a? Then only one triangle is formed since b is too long to create a second triangle on this side of the angle measure, beta.
Understanding the ambiguous case of the Law of Sines can be difficult at first. I recommend practicing with the included Geogebra file (http://ggbtu.be/m144368 ). In it, you can modify the lengths of each side and the excluded angle to see what kind of triangles can be formed. While practicing, be sure to calculate your h values as well so that you can see the relation between h and the types of triangles formed. Good luck!
Question
How does finding the height of a given triangle help us determine the ambiguous case of the Law of Sines?
