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8-6 The Law of S ines and Law of Cosines

8-6 The Law of S ines and Law of Cosines. You used trigonometric ratios to solve right triangles. Use the Law of Sines to solve triangles. Use the Law of Cosines to solve triangles. Law of Sines.

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8-6 The Law of S ines and Law of Cosines

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  1. 8-6 The Law of Sines and Law of Cosines You used trigonometric ratios to solve right triangles. • Use the Law of Sines to solve triangles. • Use the Law of Cosines to solve triangles.

  2. Law of Sines • The Law of Sines can be used to find side lengths and angle measures for any triangle (nonright triangles). • You can use the Law of Sines to solve a triangle if you know the measures of two angles and any side (AAS or ASA) p. 588

  3. Law of Sines (AAS or ASA) Divide each side by sin Find p. Round to the nearest tenth. We are given measures of two angles and a nonincluded side, so use the Law of Sines to write a proportion (AAS). Law of Sines Cross Products Property Use a calculator. Answer:p≈ 4.8

  4. Find c to the nearest tenth. A. 4.6 B. 29.9 C. 7.8 D. 8.5

  5. 6 57° x Law of Sines (ASA) Find x. Round to the nearest tenth. Law of Sines mB = 50, mC = 73, c = 6 6 sin 50 = x sin 73 Cross Products Property Divide each side by sin 73. 4.8 = x Use a calculator. Answer:x ≈ 4.8

  6. x 43° Find x. Round to the nearest degree. A. 8 B. 10 C. 12 D. 14

  7. Law of Cosines You can use the Law of Cosines to solve a triangle if you know the measures of two sides and the included angle (SAS) p. 589

  8. Law of Cosines (SAS) Find x. Round to the nearest tenth. Use the Law of Cosines since the measures of two sides and the included angle are known. Law of Cosines Simplify. Take the square root of each side. Use a calculator. Answer:x ≈ 18.9

  9. Find r if s = 15, t = 32, and mR = 40. Round to the nearest tenth. A. 25.1 B. 44.5 C. 22.7 D. 21.1

  10. Law of Cosines (SSS) Find mL. Round to the nearest degree. Law of Cosines Simplify. Subtract 754 from each side. Divide each side by –270. Solve for L. Use a calculator. Answer:mL≈ 49

  11. Find mP. Round to the nearest degree. A. 44° B. 51° C. 56° D. 69°

  12. 8-6 Assignment, day 1 p. 592, 12-17, 22-27

  13. 8-6 The Law of Sines and Law of Cosines, day 2 You used trigonometric ratios to solve right triangles. • Use the Law of Sines to solve triangles. • Use the Law of Cosines to solve triangles.

  14. Solve a Triangle When solving right triangles, you can use sine, cosine, or tangent. When solving other triangles, you can use the Law of Sinesor the Law of Cosines, depending on what information is given AAS, ASA for sines SAS, SSS for cosines)

  15. AIRCRAFT From the diagram of the plane shown, determine the approximate width of each wing. Round to the nearest tenth meter. Use the Law of Sines to find KJ. Law of Sines Cross products

  16. Divide each side by sin . Cross products Simplify. Answer:The width of each wing is about 16.9 meters.

  17. The rear side window of a station wagon has the shape shown in the figure. Find the perimeter of the window if the length of DB is 31 inches. Round to the nearest tenth. A. 93.5 in. B. 103.5 in. C. 96.7 in. D. 88.8 in.

  18. Solve a Triangle Solve triangle PQR. Round to the nearest degree. Since the measures of three sides are given (SSS), use the Law of Cosines to find mP. p2 = r2 + q2 – 2pqcosP Law of Cosines 82 = 92 + 72 – 2(9)(7) cosP p = 8, r = 9, and q = 7 64= 130 – 126 cosP Simplify. –66= –126 cosP Subtract 130 fromeach side. Divide each side by –126. Use the inverse cosine ratio. Use a calculator.

  19. Use the Law of Sines to find mQ. Law of Sines mP≈ 58, p = 8,q = 7 Multiply each side by 7. Use the inverse sine ratio. Use a calculator. By the Triangle Angle Sum Theorem, mR≈ 180 – (58 + 48) or 74. Answer:Therefore, mP≈ 58; mQ≈ 48 andmR≈ 74.

  20. Solve ΔRST. Round to the nearest degree. A.mR = 82, mS = 58, mT = 40 B.mR = 58, mS = 82, mT = 40 C.mR = 82, mS = 40, mT = 58 D.mR = 40, mS = 58, mT = 82

  21. SAS AAS ASA p. 592

  22. 8-6 Assignment day 2 p. 592, 31-42

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