CK-12 Trigonometry
Page 21
Use the Law of Sines in real-world and applied problems.
Introduction
Real-World Application:
Consider an airline flight: In order to avoid a large and dangerous snowstorm on a flight from Chicago to Buffalo, pilot John starts out off of the normal flight path. After flying in this direction, he turns the plane toward Buffalo. The angle formed by the first flight course and the second flight course is .
Neither the Theorem of Pythagoras or the Law of Cosines applies here. We are given two angles and a side, but the side is not included. For the pilot, two issues are pressing:
What is the total distance of the modified flight path?
How much further did he travel than if he had stayed on course?
We will solve this problem later on.
By now, we have learned that the Theorem of Pythagoras and trigonometry functions such as sine, cosine, and tangent are useful when we need to find a missing angle or side in a right triangle. But what do we use when we want to find sides or angles in triangles that are not right triangles?
We also learned about the Law of Cosines, a generalization of the Theorem of Pythagoras for non-right triangles, in a previous lesson. We know that we can use the Law of Cosines when:
We know two sides of a triangle and the included angle (SAS) or
We know all three sides of the triangle (SSS)
What happens if the triangle we are working with doesn’t fit either of those scenarios? For example, in at the right:
The triangle is not a right triangle, which means we cannot use the Theorem of Pythagoras.
We know a side and two angles, which doesn’t fulfill the requirements for using the Law of Cosines.
This is why we need the Law of Sines.
The Law of Sines is a statement about the relationship between the sides and the angles in any triangle. While the Law of Sines will yield one correct answer in many situations, there are times when it is ambiguous, meaning that it can produce more than one answer. We will explore the ambiguity of the Law of Sines in the next section.
We can use the Law of Sines when:
We know two angles and a non-included side (AAS) or
We know two angles and the included side (ASA)
In this lesson, we will learn more about the Law of Sines and how and when we can use it. We will also look at applications of the Law of Sines, and how it can be useful in finding heights and distances when they cannot be easily measured or an uneven surface makes the measurements unreliable.
Derive Two Forms of the Law of Sines
contains altitude , which extends from and intersects . We will refer to the length of altitude as .
Or, if we divide both sides by instead:
Using the same principles, we arrive at both forms of the Law of Sines:
Form 1: (sines over sides)
Form 2: (sides over sines)
AAS (Angle-Angle-Side)
One case where we need to use the Law of Sines is when we know two of the angles in a triangle and a non-included side (AAS).
For instance, in :
We know and either or
We know and either or
We know and either or
Using the Law of Sines allows us to find the other non-included side. First we will look at how to use the Law of Sines. Then we will apply this case to a situation involving a basketball game.
Example 1:
Using above, , and . Find .
Since we know two angles and one non-included side , we can find the other non-included side .
Real-World Application:
A business group wants to build a golf course on a plot of land that was once a farm. The deed to the land is old and information about the land is incomplete. If is known to be , is known to be , is known to be , is known to be , is known to be , and is known to be , what are the lengths of the sides of each triangular piece of land? What is the area of the land?
Solution: Before we can figure out the area of the land, we need to figure out the length of each side. In triangle , we know two angles and a non-included side. This is the AAS case. First, we will find the third angle in triangle by using the Triangle Sum Theorem. Then, we will use the Law of Sines to find both and .
Now, we will find using the Law of Sines.
Next, we will find the missing side lengths in triangle . In this triangle, we again know two angles and a non-included side (AAS), which means we can use the Law of Sines.
Since both and measure , triangle is an isosceles triangle. This means that since is , is also . All we have left to find now is .
Finally, we need to calculate the area of each triangle and then add the two areas together to get the total area. From the last section, we learned two area formulas, and Heron’s Formula. In this case, since we have enough information to use either formula, we will use since it is less computationally intense.
First, we will find the area of triangle .
Next, we will find the area of triangle .
The total area is .
Answer: and the total area is .
ASA (Angle-Side-Angle)
The second case where we need to use the Law of Sines is when we know two angles in a triangle and the included side (ASA). We will begin by looking at how to use the Law of Sines to solve this case and then we will solve the Real-World Application #1, involving the flight path of a plane, from earlier.
For instance, in :
We know and
We know and
We know and
In this case, the Law of Sines allows us to find either of the non-included sides ( or ).
Example 2:
In , and . Find the measure of .
Since we know two angles and the included side we can find either of the non-included sides using the Law of Sines.
First, since we already know two of the angles in the triangle, we can find the third angle using the fact that the sum of all of the angles in a triangle must equal .
Now that we know , we can use the Law of Sines to find .
We could use a similar process to find side .
We will now refer back to Real-World Application at the beginning of the section.
Part 1: In order to find the total distance of the modified flight path, we need to know side . To find side , we will need to use the Law of Sines. Since we know two angles and the included side, this is an ASA case. Remember that in the ASA case, we need to first find the third angle in the triangle.
Answer: The total distance of the modified flight path is .
Part 2: To find how much further John had to travel, we need to know the distance of the original flight path . We can use the Law of Sines again to find .
Answer: John had to travel further.
Applications
The Law of Sines can be applied in many ways. Below are some examples of the different ways and situations to which we may apply the Law of Sines. In many ways, the Law of Sines is much easier to use than the Law of Cosines since there is much less computation involved.
Situation #1: Using the Law of Sines in conjunction with the Law of Cosines.
In the figure at the right, and . Find .
First, in order to find , we must know one side in . In , we know two sides and an angle, which means we can use the Law of Cosines to find . In this case, we will refer to side as .
Now that we know , we can use the Law of Sines to find . In this case, we will refer to as .
Answer:
Situation #2: Using the Law of Sines in Conjunction with trigonometry functions.
Real-World Application: A group of forest rangers are hiking through Denali National Park towards Mt. McKinley, the tallest mountain in North America. From their campsite, they can see Mt. McKinley, and the angle of elevation from their campsite to the summit is . They know that the slope of mountain forms a angle with ground and that the vertical height of Mt. McKinley is How far away is their campsite from the base of the mountain? If they c
an hike in an hour, how long will it take them to get the base?
As you can see from the figure above, we have two triangles to deal with here: a right triangle and non-right triangle . In order to find the distance from the campsite to the base of the mountain we first need to know one side of our non-right triangle, .
If we look at in , we can see that side is our opposite side and side in our hypotenuse. Remember that the sine function is the .
Therefore we can find side using the sine function.
Now that we know side , we know two angles and the non-included side in . We can use the Law of Sines to solve for side .
If they can hike per hour:
Answer: Their campsite is approximately from the base of the mountain and it will take them about and to hike there.
Points to Consider
Are there any situations where we might not be able to use the Law of Sines or the Law of Cosines?
Considering what you already know about the sine function, is it possible for two angles to have the same sine? How might this affect using the Law of Sines to solve for an angle?
By using both the Law of Sines and the Law of Cosines, it is possible to solve any triangle we are given?
Lesson Summary
The Law of Sines has two forms:
There are two cases where we use the Law of Sines:
AAS (angle-angle-side)
ASA (angle-side-angle)
The AAS case allows us to find the other non-included side.
The ASA case allows us to find either of the non-included sides.
We can use the Triangle Sum Theorem to find the third angle in either of these cases.
The Law of Sines can be applied to different real-world situations. We’ve already explored three different situations where the Law of Sines can be applied. We will look at more situations in the Review Questions.
Review Questions
In the table below, you are given a figure and information known about that figure. Decide if each situation represents the AAS case or the ASA case.
Given Figure Case
a.
b.
c.
d.
e.
f.
Even though the triangles and given information in the table above represent two different cases of the Law of Sines, what do they all have in common?
Using the figures and the given information from the table above, find the following if possible: side
side
side
side
side
side
In , , and . Find and .
Use the Law of Sines to show that is true.
For each figure below, state whether you would use the Law of Sines, the Law of Cosines, or the one of the trig functions to solve for .
Use the Law of Sines, the Law of Cosines, and trigonometry functions to solve for .
In order to avoid a storm, a pilot starts out off path. After he has flown , he turns the plane toward his destination. The angle formed between his first path and his second path is . If the plane traveled at an average speed of per hour, how much longer did the modified flight take?
A delivery truck driver has three stops to make before she must return to the warehouse to pick up more packages. The warehouse, Stop , and Stop are all on First Street. Stop is on the corner of First Street and Route , which intersect at a angle. Stop is on the corner of First Street and Main Street, which intersect at a angle. Stop is at the intersection of Main Street and Route . The driver knows that Stop and Stop are apart and that the warehouse is from Stop . If she must be back to the warehouse by 10:00 a.m., travels at a speed of , and takes to deliver each package, at what time must she leave?
A surveyor has the job of determining the distance across the Palo Duro Canyon in Amarillo, Texas, the second largest canyon in the United States. Standing on one side of the canyon, he measures the angle formed by the edge of the canyon and the line of sight to a large boulder on the other side of the canyon. He then walks and measures the angle formed by the edge of the canyon and the new line of sight to the boulder. If the first angle formed is and the second angle formed is , find the distance across the canyon.
The surveyor spots another boulder while he is at his second spot, and finds that it forms a angle with his line of sight. He then walks further and finds that the boulder forms a angle with this line of sight. What is the distance between the two boulders?
Review Answers
ASA
AAS
neither
ASA
AAS
AAS
Student answers will vary but they should notice that in both cases you know or can find an angle and the side across from it.
not enough information
Side and side
Law of Cosines
Tangent function
Law of Sines or Cosines
Law of Sines
The modified flight took longer.
The driver must leave by 8:49 a.m.
The distance across the canyon is .
The distance between the two boulders is .
Supplemental Links
PowerPoint presentation on the Law of Sines: http://www.mente.elac.org/presentations/law_sines.pps
Vocabulary
included angle
The angle in between two known sides of a triangle.
included side
The side in between two known sides of a triangle.
Law of Sines
A statement about the relationship between the sides and the angles in any triangle.
non-included angle
An angle that is not in between two known sides of a triangle.
non-included side
A side that is not in between two known sides of a triangle.
The Ambiguous Case
Learning Objectives
A student will be able to:
Find possible triangles given two sides and an angle (SSA).
Use the Law of Cosines in various ambiguous cases.
Use the Law of Sines in various ambiguous cases.
Apply the Law of Sines and Cosines to real-world and applied problems involving the ambiguous case.
Introduction
Real-World Application: A boat leaves lighthouse and travels It is spotted from lighthouse , which is away from lighthouse . The boat forms an angle of with both lighthouses. How far is the boat from lighthouse ?
In the example above, we are given two sides of a triangle and a non-included angle (SSA). This is a case that we have not yet encountered. We will refer back to this example later on.
In previous sections, we learned about the Law of Cosines and the Law of Sines. We learned that we can use the Law of Cosines when
we know all three sides of a triangle (SSS) and
we know two sides and the included angle (SAS).
We learned that we can use the Law of Sines when
we know two angles and a non-included side (AAS) and
we know two angles and the included side (ASA).
However, we have not explored how to approach a triangle when we know two sides and a non-included angle (SSA). In this section, we will look at why the SSA case is called the ambiguous case, the possible triangles formed by the SSA case, and how to apply the Law of Sines and the Law of Cosines when we encounter the SSA case.
Possible Triangles Given SSA
In Geometry, you learned that two sides and a non-included angle do not necessarily define a unique triangle. Consider the following cases given and
Case 1: No triangle exists
In this case and side is too short to reach the base of the triangle. Since no triangle exists, there is no solution.
Case 2: One triangle exists
In this case, and side is perpendicular to the base of the triangle. Since this situation yields exactly one triangle, there is exactly one solution.
Case 3: Two triangles exist
In this case,
and side a meets the base at exactly two points. Since two triangles exist, there are two solutions. This is referred to as the ambiguous case.
Case 4: One triangle exists
In this case and side a meets the base at exactly one point. Since there is exactly one triangle, there is one solution.
Case 5: One triangle exists
In this case, and side a meets the base at exactly one point. Since there is exactly one triangle, there is one solution.
Case 3 is referred to as the Ambiguous case because there are two possible triangles and two possible solutions. One way to check to see how many possible solutions (if any) a triangle will have is to compare sides and .
If you are faced with the first situation, where , we can still tell how many solutions there will be by using and .
In the next two sections we will look at how to use the Law of Cosines and the Law of Sines when faced with the various cases above.