1. The Law of Cosines Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT TOPICSBACKNEXT Click one of the buttons below or press the enter key © 2002 East Los Angeles College. All rights reserved.

2. Topics Equations General Strategies for Using the Law of Cosines SAS SSS Click on the topic that you wish to view... EXIT BACKNEXTTOPICS

3. When solving an oblique triangle, using one of three available equations utilizing the cosine of an angle is handy. The equations are as follows: EXIT BACKNEXTTOPICS

4. EXIT BACKNEXTTOPICS

5. Note: The angle opposite a in equation 1 is . The angle opposite b in equation 2 is . The angle opposite c in equation 3 is . EXIT BACKNEXTTOPICS

6. Where did these three equations come from? EXIT BACKNEXTTOPICS

7. Create an altitude h. EXIT BACKNEXTTOPICS

8. We’ve split our original oblique triangle into two triangles. First TriangleSecond Triangle EXIT BACKNEXTTOPICS

9. First TriangleSecond Triangle EXIT BACKNEXTTOPICS

10. Our picture becomes: EXIT BACKNEXTTOPICS

11. Note the base of our triangles. First TriangleSecond Triangle adj EXIT BACKNEXTTOPICS

12. Our triangles now become, EXIT BACKNEXTTOPICS

13. *Consider two important relationships: EXIT BACKNEXTTOPICS

14. Using Relationship 1, we obtain: EXIT BACKNEXTTOPICS

15. Take a closer look at Relationship 2. EXIT BACKNEXTTOPICS

16. We now have, EXIT BACKNEXTTOPICS

17. Now, by the Pythagorean Theorem, First Triangle EXIT BACKNEXTTOPICS

18. Second Triangle EXIT BACKNEXTTOPICS

19. Why don’t you try the third equation. EXIT BACKNEXTTOPICS

20. General Strategies for Using the Law of Cosines EXIT BACKNEXTTOPICS

21. The formula for the Law of Cosines makes use of three sides and the angle opposite one of those sides. We can use the Law of Cosines: a. if we know two sides and the included angle, or b. if we know all three sides of a triangle. EXIT BACKNEXTTOPICS

22. Two sides and one angles are known. SAS EXIT BACKNEXTTOPICS

23. 87.0° 15.017.0 c   From the model, we need to determine c, , and . We start by applying the law of cosines. SAS EXIT BACKNEXTTOPICS

24. To solve for the missing side in this model, we use the form: In this form,  is the angle between a and b, and c is the side opposite . 87.0° 15.017.0 c   a b EXIT BACKNEXTTOPICS

25. Using the relationship c 2 = a 2 + b 2 – 2ab cos  We get c 2 = 15.0 2 + 17.0 2 – 2(15.0)(17.0)cos 89.0° = 225 + 289 – 510(0.0175) = 505.10 Soc = 22.5 EXIT BACKNEXTTOPICS

26. Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines to complete the problem. and This gives and Note that due to round-off error, the angles do not add up to exactly 180°. EXIT BACKNEXTTOPICS

27. Three sides are known. SSS EXIT BACKNEXTTOPICS

28. SSS 31.4 23.2 38.6 In this figure, we need to find the three angles, , , and . EXIT BACKNEXTTOPICS

29. To solve a triangle when all three sides are known we must first find one angle using the Law of Cosines. We must isolate and solve for the cosine of the angle we are seeking, then use the inverse cosine to find the angle. EXIT BACKNEXTTOPICS

30. We do this by rewriting the Law of Cosines equation to the following form: In this form, the square being subtracted is the square of the side opposite the angle we are looking for. 31.423.2 38.6 Angle to look for Side to square and subtract EXIT BACKNEXTTOPICS

31. We start by finding cos . 31.4 23.2 38.6 EXIT BACKNEXTTOPICS

32. From the equation we get and EXIT BACKNEXTTOPICS

33. 31.423.2 38.6 36.9° Our triangle now looks like this: Again, since we have the measure for both a side and the angle opposite it, we can use the Law of Sines to complete the solution of this triangle. EXIT BACKNEXTTOPICS

34. 31.4 23.2 38.6 36.9° Completing the solution we get the following: and EXIT BACKNEXTTOPICS

35. Solving these two equations we get the following: and Again, because of round-off error, the angles do not add up to exactly 180 . EXIT BACKNEXTTOPICS

36. Most of the round-off error can be avoided by storing the exact value you get for  and using that value to compute sin . Then, store sin  in your calculator’s memory also and use that value to get  and . EXIT BACKNEXTTOPICS

37. In this case we get the following: If we round off at this point we get  = 36.9°,  = 54.4° and  = 88.7°. Now the three angles add up to 180°. EXIT BACKNEXTTOPICS

38. End of Law of Cosines Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA 91754 Phone: (323) 265-8784 Fax: (323) 415-4108 Email Us At: menteprog@hotmail.com Our Website: http://www.matematicamente.org EXIT BACKNEXTTOPICS