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Trigonometry can help us solve non-right triangles as well. Non-right triangles are know as oblique triangles. There are two categories of oblique triangles—acute and obtuse. TOPICS. BACK. NEXT. EXIT. Acute Triangles. In an acute triangle, each of the angles is less than 90 º. TOPICS.

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  1. Trigonometry can help us solve non-right triangles as well. Non-right triangles are know as oblique triangles. There are two categories of oblique triangles—acute and obtuse. TOPICS BACK NEXT EXIT

  2. Acute Triangles In an acute triangle, each of the angles is less than 90º. TOPICS BACK NEXT EXIT

  3. Obtuse Triangles In an obtuse triangle, one of the angles is obtuse (between 90º and 180º). Can there be two obtuse angles in a triangle? TOPICS BACK NEXT EXIT

  4. The Law of Sines TOPICS BACK NEXT EXIT

  5. Consider the first category, an acute triangle (, ,  are acute). TOPICS BACK NEXT EXIT

  6. Create an altitude, h. * * * * TOPICS BACK NEXT EXIT

  7. Let’s create another altitude h’. TOPICS BACK NEXT EXIT

  8. * * * * TOPICS BACK NEXT EXIT

  9. Putting these together, we get This is known as the Law of Sines. TOPICS BACK NEXT EXIT

  10. The Law of Sines is used when we know any two angles and one side or when we know two sides and an angle opposite one of those sides. TOPICS BACK NEXT EXIT

  11. Fact The law of sines also works for oblique triangles that contain an obtuse angle (angle between 90º and 180º).  is obtuse TOPICS BACK NEXT EXIT

  12. General Strategies for Usingthe Law of Sines TOPICS BACK NEXT EXIT

  13. One side and two angles are known. ASA or SAA TOPICS BACK NEXT EXIT

  14. ASA From the model, we need to determine a, b, and  using the law of sines. TOPICS BACK NEXT EXIT

  15. First off, 42º + 61º +  = 180º so that  = 77º. (Knowledge of two angles yields the third!) TOPICS BACK NEXT EXIT

  16. Now by the law of sines, we have the following relationships: TOPICS BACK NEXT EXIT

  17. So that ● ● ● ● TOPICS BACK NEXT EXIT

  18. SAA From the model, we need to determine a, b, and  using the law of sines.Note: + 110º + 40º = 180º so that  = 30º b a TOPICS BACK NEXT EXIT

  19. By the law of sines, TOPICS BACK NEXT EXIT

  20. Thus, ● ● ● ● TOPICS BACK NEXT EXIT

  21. The Ambiguous Case – SSA In this case, you may have information that results in one triangle, two triangles, or no triangles. TOPICS BACK NEXT EXIT

  22. SSA – No Solution Two sides and an angle opposite one of the sides. TOPICS BACK NEXT EXIT

  23. By the law of sines, TOPICS BACK NEXT EXIT

  24. Thus, Therefore, there is no value for  that exists! No Solution! TOPICS BACK NEXT EXIT

  25. SSA – Two Solutions TOPICS BACK NEXT EXIT

  26. By the law of sines, TOPICS BACK NEXT EXIT

  27. So that, TOPICS BACK NEXT EXIT

  28. Case 1Case 2 Both triangles are valid! Therefore, we have two solutions. TOPICS BACK NEXT EXIT

  29. Case 1 * * TOPICS BACK NEXT EXIT

  30. Case 2 * * TOPICS BACK NEXT EXIT

  31. Finally our two solutions: TOPICS BACK NEXT EXIT

  32. SSA – One Solution TOPICS BACK NEXT EXIT

  33. By the law of sines, TOPICS BACK NEXT EXIT

  34. * * TOPICS BACK NEXT EXIT

  35. Note– Only one is legitimate! TOPICS BACK NEXT EXIT

  36. Thus we have only one triangle. TOPICS BACK NEXT EXIT

  37. By the law of sines, * * TOPICS BACK NEXT EXIT

  38. Finally, we have: TOPICS BACK NEXT EXIT

  39. End of Law of SinesHomework – Pg 484 1-11 odd, 13-15, 19, 23, 27, 28, 37-39, 45 TOPICS BACK NEXT EXIT

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