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Word Problems: Law of Sines
In order to solve problems which require the application of the Law of Sines, it is necessary to
 
 
A typical problem requiring the Law of Sines in order to solve it involves a triangle in which there is no right angle. We are given some information about a triangle, but we have to find measurements of other sides and/or angles. The Law of Sines for a triangle ABC is stated below, assuming that the side opposite angle A is a, the side opposite angle B is b, and the side opposite angle C is c:
 
 
Suppose in triangle ABC that are given. Find the measure of side c. This would be a typical example of this type of problem.
 
First, we make a diagram. A diagram of this triangle is shown below.
 
 
In this diagram the given distances and angles are labeled:
 
 
 
The variable c is chosen to represent the unknown measurement of the side opposite angle C. This is the object of the question.
 
To relate the known measurements and the variable, an equation is written. In this case the equation involves the ratios of the sines of angles to the opposite sides. We have
 
 
We now need to know the measure of angle B to solve the problem.
 
The sum of angles A and C is 28º + 91º = 119º. Since the sum of the angles in a triangle equals 180º  we know that angle B must have a measure of
 
.
 
Therefore the equation we use to find side c is
 
.
 
This equation give us
 
.
 

Let's Practice

Question #1
AudioJohn wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33º. This particular tree grows at an angle of 83º with respect to the ground rather than vertically (90º). How tall is the tree?


Question #2
Audio
A building is of unknown height. At a distance of 100 feet away from the building, an observer notices that the angle of elevation to the top of the building is 41º and that the angle of elevation to a poster on the side of the building is 21º. How far is the poster from the roof of the building?



Try These
Question #1
Audio
Triangle ABC has , , and side a = 42.9 inches. What is the measure of side c?


Question #2
Audio
An observer is near a river and wants to calculate the distance across the river. He measures the angle between his observations of two points on the shore, one on his side and one on the other side, to be 28º. The distance between him and the point on his side of the river can be measured and is 300 feet. The angle formed by him, the point on his side of the river, and the point directly on the opposite side of the river is 128º. What is the distance across the river?



As you can see, this type of problem requires a diagram of a carefully labeled triangle. The measurements of a side and two angles should be labeled based on the information given. One measurement of a side in the triangle is missing. It is the goal of the problem to find this measurement. First you should assign a variable to represent the missing measurement. We usually use a lower case English letter to represent the measure of a side of the triangle.
 
Use of the Law of Sines involves a simple equation. It is important to set a calculator for degrees if that is the manner in which the angles are measured. If one angle and two sides are known, it is best to use the Law of Cosines to find the measurements of missing parts of the triangle.

M Ransom

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