1. H.Melikian/12001 7.2 Cosine Law and Area of Triangle Dr.Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu 1. Determining if the Law of Sines or the Law of Cosines Should be Used to Begin to Solve an Oblique Triangle 2. Using the Law of Cosines to Solve the SAS Case 3. Using the Law of Cosines to Solve the SSS Case 4. Using the Law of Cosines to Solve Applied Problems Involving Oblique Triangles 5. Determining the Area of Oblique Triangles 6. Using Heron’s Formula to Determine the Area of an SSS Triangle 7. Solving Applied Problems Involving the Area of Triangles

2. H.Melikian/12002 The Law of Cosines If A, B, and C are the measures of the angles of any triangle and if a, b, and c are the lengths of the sides opposite the corresponding angles, then

3. H.Melikian/12003 Determining of the Law of Sines or Cosines Should Be Used to Begin to Solve an Oblique Triangle Decide whether the Law of Sines or Cosines can be used to solve each triangle. Do not solve. a.b. Law of Sines; given the lengths of two sides and an angle opposite one of the given sides. Neither can be used.

4. H.Melikian/12004 Solving an SAS Oblique Triangle Step 1 : Use the Law of Cosines to determine the length of the missing side. Step 2: Determine the measure of the smaller of the remaining two angles using the Law of Sines or using the alternate form of the Law of Cosines. Step 3: Use the fact that the sum of the measures of the three angles of a triangle is 180 degrees to determine the measure of the remaining angle.

5. H.Melikian/12005 Solving a SAS Triangle Solve the given oblique triangle. Round all measurements to one decimal place. AnglesSides A = 61 a = B = b = 20 C = c = 30

6. H.Melikian/12006 Solving a SAS Triangle-cont Solve the given oblique triangle. Round all measurements to one decimal place. AnglesSides A = 61 a = B = b = 20 C = c = 30

7. H.Melikian/12007 Solving an SSS Oblique Triangle Step 1: Use the alternate form of the Law of Cosines to determine the length of the largest angle. This is the angle opposite the longest side. Step 2: Determine the measure of one of the remaining two angles using the Law of Sines or the alternate form of the Law of Cosines. Step 3: Use the fact that the sum of the measures of the three angles of a triangle is 180 degrees to determine the measure of the remaining angle.

8. H.Melikian/12008 Solving a SSS Triangle Solve oblique triangle ABC if a = 4, b = 3, and c = 6. AnglesSides A = a = 4 B = b = 3 C = c = 6

9. H.Melikian/12009 Solving a SSS Triangle-cont Solve oblique triangle ABC if a = 4, b = 3, and c = 6. AnglesSides A = a = 4 B = b = 3 C = c = 6 153.6 degrees will not work.

10. H.Melikian/120010 Determining the Distance between Two Airplanes Two planes take off from different runways at the same time. One plane flies at an average speed of 350 mph with a bearing of N 21  E. The other plane flies at an average speed of 420 mph with a bearing of S 84  W. How far apart are the planes from each other 2 hours after takeoff?

11. H.Melikian/120011 Determining the Distance between Two Airplanes Two planes take off from different runways at the same time. One plane flies at an average speed of 350 mph with a bearing of N 21  E. The other plane flies at an average speed of 420 mph with a bearing of S 84  W. How far apart are the planes from each other 2 hours after takeoff? a = 700, b = 840, and D = 117  The planes are about 1315 miles apart after 2 hours.

12. H.Melikian/120012 Area of a Triangle In any triangle, the area is given by where b is the length of the base of the triangle and h is the length of the altitude drawn to that base (or drawn to an extension of that base).

13. H.Melikian/120013 Area of a Triangle If A, B, and C are the measures of the angles of any triangle and if a, b, and c are the lengths of the sides opposite the corresponding angles, then the area of triangle ABC is given by

14. H.Melikian/120014 Determining the Area of an Oblique Triangle Determine the area of the triangle.

15. H.Melikian/120015 Determining the Area of an Oblique Triangle Determine the area of the triangle.

16. H.Melikian/120016 Heron’s Formula Suppose that a triangle has side lengths of a, b, and c. If the semiperimeter is then the area of the triangle is

17. H.Melikian/120017 Determining the Area of an SSS Oblique Triangle Using Heron’s Formula Determine the area of the triangle.