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Law of Cosines

            The Law of Cosines can be used to solve a triangle when either (1)two sides and the included angle are known, or (2)three sides are known. A statement of the Law of Cosines is given below:

 

            For any triangle with sides a, b, and c and opposite angles

, respectively,

                                                

            Letís look at an example that shows how to use the Law of Cosines.

Example

Given two sides of a triangle  and the angle  find the remaining parts of the triangle.

Solution

 

                

 

   We can use the Law of Cosines to find the missing side c. See figure 1.

         

      

                   

We can use the Law of Cosines again to find the missing angles:

For a,

For b,

Notice that , which is what we would expect (except for rounding error) since the sum of the measures of the angles in a triangle should always be 180į.