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Geometry Law of Sines and Law of Cosines

Geometry Law of Sines and Law of Cosines. Warm up. Use the given trigonometric ratio to determine which angle of the triangle is /A. cos A = 15/17 sin A = 15/17 tan A = 1.875. Law of Sines and Law of Cosines.

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Geometry Law of Sines and Law of Cosines

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  1. Geometry Law of Sines and Law of Cosines CONFIDENTIAL

  2. Warm up Use the given trigonometric ratio to determine which angle of the triangle is /A. • cos A = 15/17 • sin A = 15/17 • tan A = 1.875 CONFIDENTIAL

  3. Law of Sines and Law of Cosines In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to 180˚. CONFIDENTIAL

  4. Finding Trigonometric Ratios for Obtuse Angles Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. A) sin 135˚ B) tan 98 ˚ C) cos 108 ˚ CONFIDENTIAL

  5. Now you try! 1) Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. a) tan 175˚ b) cos 92˚ c) sin 160˚ CONFIDENTIAL

  6. You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In ∆ABC, let h represent the length of the altitude from C to AB. From the diagram, sin A = (h), and sin B = (h). By solving for h, you find that h = bsin A And h = a.sinB .So b sin A = a.sinB, And (sin A/a) = (sin B/b). You can use another altitude to show that these ratios Equal (sin C/c). ‾ b a CONFIDENTIAL

  7. The Law of Sines Theorem 1.1 CONFIDENTIAL

  8. You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) or • two side length and a non-include angle measure (SSA). CONFIDENTIAL

  9. Use the Law of Sines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. Law of Sines Substitute the given values. Cross Products Property Divide both sides by sin 105˚. Next page  CONFIDENTIAL

  10. Law of Sines Substitute the given values. Multiply both sides by 5. Use the inverse sine function to find m/s. CONFIDENTIAL

  11. Now you try! 2) Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. CONFIDENTIAL

  12. The Law of Sine cannot be used to solve every triangle. If you know two side lengths and the include angle measure or if you know all three side lengths. You cannot use the Law of Sines.Instead, you can apply the Law of Cosines. CONFIDENTIAL

  13. The Law of Cosines Theorem 1.2 CONFIDENTIAL

  14. You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included angle measure (SAS) or • three side lengths (SSS). CONFIDENTIAL

  15. Using the Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Next page  CONFIDENTIAL

  16. Law of Cosines Substitute the given values. Simplify Subtract 65 from both sides. Solve for cos R Use the inverse Cosine function to find m/R. CONFIDENTIAL

  17. Now you try! 3) Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. CONFIDENTIAL

  18. Engineering Application The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100˚ angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Next page  CONFIDENTIAL

  19. Law of Sines Substitute the Calculated values. Multiply both sides by 56. Use the inverse sine function to find m/A. CONFIDENTIAL

  20. Now you try! 4) Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. CONFIDENTIAL

  21. Now some practice problems for you! CONFIDENTIAL

  22. Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. • 1) sin 100˚ 2) cos 167˚ 3) tan 92˚ • 4) tan 141˚ 5) sin 150˚ 6) cos 133˚ CONFIDENTIAL

  23. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 7) CONFIDENTIAL

  24. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 8) 9) CONFIDENTIAL

  25. 10) A carpenter makes a triangular frame by joining three pieces of wood that are 20 cm, 24 cm, and 30 cm long. What are the measures of the angles of the triangle? Round to the nearest degree. CONFIDENTIAL

  26. Let’s review Finding Trigonometric Ratios for Obtuse Angles Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. A sin 135˚ B tan 98 ˚ C cos 108 ˚ CONFIDENTIAL

  27. You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In ∆ABC, let h represent the length of the altitude from C to AB. From the diagram, sin A = (h/b), and sin B = (h/a). By solving for h, you find that h = bsin A And h = asin B .So b sin A = asin B, And (sin A/a) = (sin B/b). You can use another altitude to show that these ratios Equal (sin C/c). ‾ CONFIDENTIAL

  28. The Law of Sines Theorem 1.1 CONFIDENTIAL

  29. You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) or • two side length and a non-include angle measure (SSA). CONFIDENTIAL

  30. Use the Law of Sines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. Law of Sines Substitute the given values. Cross Products Property Divide both sides by sin 105˚. Next page  CONFIDENTIAL

  31. Law of Sines Substitute the given values. Multiply both sides by 5. Use the inverse sine function to find m/s. CONFIDENTIAL

  32. The Law of Sine cannot be used to solve every triangle. If you know two side lengths and the include angle measure or if you know all three side lengths. You cannot use the Law of Sines. Instead, you can apply the Law of Cosines. CONFIDENTIAL

  33. The Law of Cosines Theorem 1.2 CONFIDENTIAL

  34. You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included angle measure (SAS) or • three side lengths (SSS). CONFIDENTIAL

  35. Using the Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Next page  CONFIDENTIAL

  36. Law of Cosines Substitute the given values. Simplify Subtract 65 from both sides. Solve for cos R Use the inverse Cosine function to find m/R. CONFIDENTIAL

  37. Engineering Application The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100˚ angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Next page  CONFIDENTIAL

  38. Law of Sines Substitute the Calculated values. Multiply both sides by 56. Use the inverse sine function to find m/A. CONFIDENTIAL

  39. You did a great job today! CONFIDENTIAL

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