# Further Applications of Trigonometry

## Non-right Triangles: Law of Sines

To ensure the safety of over 5,000 U.S. aircraft flying simultaneously during peak times, air traffic controllers monitor and communicate with them after receiving data from the robust radar beacon system. Suppose two radar stations located 20 miles apart each detect an aircraft between them. The angle of elevation measured by the first station is 35 degrees, whereas the angle of elevation measured by the second station is 15 degrees. How can we determine the altitude of the aircraft? We see in  that the triangle formed by the aircraft and the two stations is not a right triangle, so we cannot use what we know about right triangles. In this section, we will find out how to solve problems involving non-right triangles.

### Using the Law of Sines to Solve Oblique Triangles

In any triangle, we can draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles. It would be preferable, however, to have methods that we can apply directly to non-right triangles without first having to create right triangles.
Any triangle that is not a right triangle is an oblique triangle. Solving an oblique triangle means finding the measurements of all three angles and all three sides. To do so, we need to start with at least three of these values, including at least one of the sides. We will investigate three possible oblique triangle problem situations:
1. ASA (angle-side-angle) We know the measurements of two angles and the included side. See .

    
2. AAS (angle-angle-side) We know the measurements of two angles and a side that is not between the known angles. See .

    
3. SSA (side-side-angle) We know the measurements of two sides and an angle that is not between the known sides. See .
    
Knowing how to approach each of these situations enables us to solve oblique triangles without having to drop a perpendicular to form two right triangles. Instead, we can use the fact that the ratio of the measurement of one of the angles to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. Let’s see how this statement is derived by considering the triangle shown in .
Using the right triangle relationships, we know that 
  and 
  Solving both equations for 
       gives two different expressions for 
      
    
We then set the expressions equal to each other.
Similarly, we can compare the other ratios.
Collectively, these relationships are called the Law of Sines.
Note the standard way of labeling triangles: angle 
       (alpha) is opposite side 
       angle 
       (beta) is opposite side 
       and angle 
       (gamma) is opposite side 
       See .
While calculating angles and sides, be sure to carry the exact values through to the final answer. Generally, final answers are rounded to the nearest tenth, unless otherwise specified.

### Using The Law of Sines to Solve SSA Triangles

We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. In some cases, more than one triangle may satisfy the given criteria, which we describe as an ambiguous case. Triangles classified as SSA, those in which we know the lengths of two sides and the measurement of the angle opposite one of the given sides, may result in one or two solutions, or even no solution.

###  Finding the Area of an Oblique Triangle Using the Sine Function

Now that we can solve a triangle for missing values, we can use some of those values and the sine function to find the area of an oblique triangle. Recall that the area formula for a triangle is given as 
       where 
       is base and 
       is height. For oblique triangles, we must find 
       before we can use the area formula. Observing the two triangles in , one acute and one obtuse, we can drop a perpendicular to represent the height and then apply the trigonometric property 
  to write an equation for area in oblique triangles. In the acute triangle, we have 
       or 
       However, in the obtuse triangle, we drop the perpendicular outside the triangle and extend the base 
       to form a right triangle. The angle used in calculation is 
       or 
      
      
Thus,
Similarly,

###  Solving Applied Problems Using the Law of Sines

The more we study trigonometric applications, the more we discover that the applications are countless. Some are flat, diagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and motion.

### Key Equations


### Key Concepts

1. The Law of Sines can be used to solve oblique triangles, which are non-right triangles.
2. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
3. There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution. See .
4. The ambiguous case arises when an oblique triangle can have different outcomes.
5. There are three possible cases that arise from SSA arrangement—a single solution, two possible solutions, and no solution. See  and .
6. The Law of Sines can be used to solve triangles with given criteria. See .
7. The general area formula for triangles translates to oblique triangles by first finding the appropriate height value. See .
8. There are many trigonometric applications. They can often be solved by first drawing a diagram of the given information and then using the appropriate equation. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, assume 
     is opposite side 
     is opposite side 
     and 
     is opposite side 
     Solve each triangle, if possible. Round each answer to the nearest tenth.
For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle 
     is opposite side 
     angle 
     is opposite side 
     and angle 
     is opposite side 
    
  
For the following exercises, assume 
     is opposite side 
     is opposite side 
     and 
     is opposite side 
     Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.
For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.
For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

### Graphical

For the following exercises, find the length of side 
     Round to the nearest tenth.
For the following exercises, find the measure of angle 
  if possible. Round to the nearest tenth.
For the following exercise, solve the triangle. Round each answer to the nearest tenth.

### Extensions


### Real-World Applications
