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Warm_Up 5

Warm_Up 5. Find each measure to the nearest tenth. 1 . m  y 2. x 3. y. 104°. ≈ 8.8. ≈ 18.3. Page 962. Review. Pg 978 39-47, 49, 51, 57-61. Law of Cosines. Triangles equal 180 degrees Used for triangles that are not right triangles (AKA Oblique Triangles)

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Warm_Up 5

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  1. Warm_Up 5 Find each measure to the nearest tenth. 1. my 2. x 3. y 104° ≈ 8.8 ≈ 18.3 13.5 Law of Sines

  2. Page 962 13.5 Law of Sines

  3. Review Pg 978 39-47, 49, 51, 57-61 13.5 Law of Sines

  4. Law of Cosines • Triangles equal 180 degrees • Used for triangles that are not right triangles (AKA Oblique Triangles) • Vertices are represented by a capital letter • Sides are represented by a lower case letter • Used to solve triangles for which side-angle-side (SAS) or side-side-side (SSS) 10.1 - Law of Cosines

  5. Law of Cosines • Capital letters represent vertices • Lowercase letters represent sides …Do you see a pattern? 10.1 - Law of Cosines

  6. Law of Cosines Draw the figure Determine what is missing (sides and angle) Plug in Law of Cosines equation Round answers to tenths and answers need to be positive Check 10.1 - Law of Cosines

  7. Example 1 8 B a c b 32.2° 5 A C Find the length of the third side. Need to find mA, mB, and c. Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 5, mC = 32.2° 10.1 - Law of Cosines

  8. Example 1 8 4.6 B 32.2° 5 a c Find the length of the third side. b c2 = a2 + b2 – 2abcosC A C c2 = (8)2 + (5)2 – 2(8)(5)cos (32.2°) c2 ≈ 21.3 c ≈ 4.6 Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 5, mC = 32.2° 10.1 - Law of Cosines

  9. Example 1 8 4.6 32.2° B Find the length of mA. 5 a c a2 ≈ b2+c2 – 2bccos A b (8)2 ≈ (5)2+(4.6)2 – 2(5)(4.6)cos A A C 64 ≈ 25+21.16– 46cos A 17.84≈ – 46cos A Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 5, mC = 32.2° 10.1 - Law of Cosines

  10. Example 1 8 4.6 B 112.8° 32.2° Find the length of mA. 5 a c 17.84≈ –46cos A A ≈ cos-1(-0.387) b A C A ≈ 112.8° -0.387≈ cos A Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 5, mC = 32.2° 10.1 - Law of Cosines

  11. Example 1 35.4° 8 4.6 B 112.8° 32.2° 5 a Find the length of mB. c b 112. 8° +mB + 32.2°  180° A C mB 35.4° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 5, mC = 32.2° 10.1 - Law of Cosines

  12. Example 2 c 21.5 mA 44.6° mB 135.4° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 16, b = 10, mC = 110°. Solve for c, mA andmB. 10.1 - Law of Cosines

  13. Your Turn c 9.3 mA 47.8° mC 32.2° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 7, c = 5, mB = 100°. Solve for c, mA andmB. 10.1 - Law of Cosines

  14. Example 3 B a c 8 7 b A C 9 Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 9, c= 7. Solve for mA, mB, and mC, 10.1 - Law of Cosines

  15. Example 3 73.4° B a Find the length of mB. c 8 7 b2= a2 + c2– 2ac cos B b A C 92= 82 + 72– 2 (8)(7)cos B 9 cosB = 0.2857 mB = cos-1 (0.2857) ≈ 73.4° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 9, c= 7. Solve for mA, mB, and mC, 10.1 - Law of Cosines

  16. Example 3 73.4° B 48.2° a Find the length of mC. c 8 7 c2= a2 + b2– 2ab cos C b A C 72= 82 + 92– 2 (8)(9)cos C 9 cosC = 0.6667 mC = cos-1 (0.6667) ≈ 48.2° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 9, c= 7. Solve for mA, mB, and mC, 10.1 - Law of Cosines

  17. Example 3 73.4° B 48.2° 58.4° a Find the length of mA. c 8 7 b A C m A +73.4° + 48.2°  180° 9 mA58.4° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 8, b = 9, c= 7. Solve for mA, mB, and mC, 10.1 - Law of Cosines

  18. Your Turn mA 43.4° mB 55.6° mC 81.0° Use the given measurements to solve ∆ABC. Round to the nearest tenth, a = 35, b = 42, c = 50.3. Solve for mA, mB, and mC. 10.1 - Law of Cosines

  19. Example 4 B 89.2° a c b A C 74.5° 16.3 ° No Solution Use the given measurements to solve ∆ABC. Round to the nearest tenth, mA = 74.5°, mB = 89.2°, mC= 16.3°. Solve for a, b, and c. 10.1 - Law of Cosines

  20. Example 5 A pilot is flying from Houston to Oklahoma City. To avoid a thunderstorm, the pilot flies 28° off the direct route for a distance of 175 miles. He then makes a turn and flies straight on to Oklahoma City. To the nearest mile, how much farther than the direct route was the route taken by the pilot? 10.1 - Law of Cosines

  21. Example 5 To the nearest mile, how much farther than the direct route was the route taken by the pilot? b2 = c2 + a2 – 2ca cos B b2 ≈ 3962 + 1752 – 2(396)(175)cos28° b2 ≈ 65072 255 + 175 = 430 b ≈ 255 430 – 396 = 34 34 MILES A pilot is flying from Houston to Oklahoma City. To avoid a thunderstorm, the pilot flies 28° off the direct route for a distance of 175 miles. He then makes a turn and flies straight on to Oklahoma City. To the nearest mile, how much farther than the direct route was the route taken by the pilot? 10.1 - Law of Cosines

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