# Further Applications of Trigonometry

## Non-right Triangles: Law of Cosines

Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown in . How far from port is the boat?
Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angle is between two known sides, a SAS (side-angle-side) triangle, or when all three sides are known, but no angles are known, a SSS (side-side-side) triangle. In this section, we will investigate another tool for solving oblique triangles described by these last two cases.

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

The tool we need to solve the problem of the boat’s distance from the port is the Law of Cosines, which defines the relationship among angle measurements and side lengths in oblique triangles. Three formulas make up the Law of Cosines. At first glance, the formulas may appear complicated because they include many variables. However, once the pattern is understood, the Law of Cosines is easier to work with than most formulas at this mathematical level.
Understanding how the Law of Cosines is derived will be helpful in using the formulas. The derivation begins with the Generalized Pythagorean Theorem, which is an extension of the Pythagorean Theorem to non-right triangles. Here is how it works: An arbitrary non-right triangle 
       is placed in the coordinate plane with vertex 
       at the origin, side 
       drawn along the x-axis, and vertex 
       located at some point 
       in the plane, as illustrated in . Generally, triangles exist anywhere in the plane, but for this explanation we will place the triangle as noted.
We can drop a perpendicular from 
       to the x-axis (this is the altitude or height). Recalling the basic trigonometric identities, we know that
In terms of 
  and 
  
 The 
       point located at 
       has coordinates 
        
           Using the side 
       as one leg of a right triangle and 
       as the second leg, we can find the length of hypotenuse 
       using the Pythagorean Theorem. Thus,
The formula derived is one of the three equations of the Law of Cosines. The other equations are found in a similar fashion.
Keep in mind that it is always helpful to sketch the triangle when solving for angles or sides. In a real-world scenario, try to draw a diagram of the situation. As more information emerges, the diagram may have to be altered. Make those alterations to the diagram and, in the end, the problem will be easier to solve.

### Solving Applied Problems Using the Law of Cosines

Just as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is applicable to situations in which the given data fits the cosine models. We may see these in the fields of navigation, surveying, astronomy, and geometry, just to name a few.

### Using Heron’s Formula to Find the Area of a Triangle

We already learned how to find the area of an oblique triangle when we know two sides and an angle. We also know the formula to find the area of a triangle using the base and the height. When we know the three sides, however, we can use Heron’s formula instead of finding the height. Heron of Alexandria was a geometer who lived during the first century A.D. He discovered a formula for finding the area of oblique triangles when three sides are known.

### Key Equations


### Key Concepts

1. The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.
2. The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be related and measurements to be calculated. See  and .
3. The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally to draw a sketch of the problem presented. If the information given fits one of the three models (the three equations), then apply the Law of Cosines to find a solution. See  and .
4. Heron’s formula allows the calculation of area in oblique triangles. All three sides must be known to apply Heron’s formula. See  and See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, assume 
         is opposite side 
          is opposite side 
         and 
          is opposite side 
         If possible, solve each triangle for the unknown side. Round to the nearest tenth.
For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.
For the following exercises, solve the triangle. Round to the nearest tenth.
For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

### Graphical

For the following exercises, find the length of side 
   Round to the nearest tenth.
For the following exercises, find the measurement of angle 
        
      
For the following exercises, solve for the unknown side. Round to the nearest tenth.
For the following exercises, find the area of the triangle. Round to the nearest hundredth.

### Extensions

For the following exercises, suppose that 
         represents the relationship of three sides of a triangle and the cosine of an angle.
For the following exercises, find the area of the triangle.

### Real-World Applications
