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The Law of Cosines

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

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The Law of Cosines

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

  2. Topics Click on the topic that you wish to view . . . EquationsGeneral Strategies for Using the Law of CosinesSASSSS TOPICS BACK NEXT EXIT

  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: TOPICS BACK NEXT EXIT

  4. TOPICS BACK NEXT EXIT

  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 . TOPICS BACK NEXT EXIT

  6. Where did these three equations come from? TOPICS BACK NEXT EXIT

  7. Create an altitude h. TOPICS BACK NEXT EXIT

  8. We’ve split our original oblique triangle into two triangles. First Triangle Second Triangle TOPICS BACK NEXT EXIT

  9. First Triangle Second Triangle TOPICS BACK NEXT EXIT

  10. Our picture becomes: TOPICS BACK NEXT EXIT

  11. Note the base of our triangles. adj adj First Triangle Second Triangle TOPICS BACK NEXT EXIT

  12. Our triangles now become, TOPICS BACK NEXT EXIT

  13. *Consider two important relationships: TOPICS BACK NEXT EXIT

  14. Using Relationship 1, we obtain: TOPICS BACK NEXT EXIT

  15. Take a closer look at Relationship 2. TOPICS BACK NEXT EXIT

  16. We now have, TOPICS BACK NEXT EXIT

  17. Now, by the Pythagorean Theorem, First Triangle TOPICS BACK NEXT EXIT

  18. Second Triangle TOPICS BACK NEXT EXIT

  19. Why don’t you try the third equation. TOPICS BACK NEXT EXIT

  20. General Strategies for Usingthe Law of Cosines TOPICS BACK NEXT EXIT

  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. TOPICS BACK NEXT EXIT

  22. Two sides and one angles are known. SAS TOPICS BACK NEXT EXIT

  23. SAS 87.0° 17.0 15.0   c From the model, we need to determine c, , and . We start by applying the law of cosines. TOPICS BACK NEXT EXIT

  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 .  a b 87.0° 17.0 15.0   c TOPICS BACK NEXT EXIT

  25. Using the relationship c2 = a2 + b2 – 2ab cos  We get c2 = 15.02 + 17.02 – 2(15.0)(17.0)cos 89.0° = 225 + 289 – 510(0.0175) = 505.10 So c = 22.5 TOPICS BACK NEXT EXIT

  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°. TOPICS BACK NEXT EXIT

  27. Three sides are known. SSS TOPICS BACK NEXT EXIT

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

  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. TOPICS BACK NEXT EXIT

  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. Side to square and subtract 31.4 23.2 Angle to look for 38.6 TOPICS BACK NEXT EXIT

  31. We start by finding cos . 23.2 31.4 38.6 TOPICS BACK NEXT EXIT

  32. From the equation we get and TOPICS BACK NEXT EXIT

  33. Our triangle now looks like this: 31.4 23.2 36.9° 38.6 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. TOPICS BACK NEXT EXIT

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

  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. TOPICS BACK NEXT EXIT

  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 . TOPICS BACK NEXT EXIT

  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°. TOPICS BACK NEXT EXIT

  38. End of Law of CosinesTitle V East Los Angeles College1301 Avenida Cesar ChavezMonterey Park, CA 91754Phone: (323) 265-8784Fax: (323) 415-4108Email Us At:menteprog@hotmail.comOur Website:http://www.matematicamente.org TOPICS BACK NEXT EXIT

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