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Understanding the Law of Sines: Area Calculation and Triangle Solving Strategies

This educational resource focuses on the Law of Sines, essential for solving triangles and calculating their areas efficiently. It explains how to find the area of a triangle using the sine function and provides practical examples for better understanding. The guide includes several problems to practice, emphasizing rounding angles to the nearest whole number before solving. Key concepts and examples are illustrated to enhance learning, making it a valuable tool for students studying trigonometry.

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Understanding the Law of Sines: Area Calculation and Triangle Solving Strategies

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  1. Merrill pg. 765, 10-18 10. 24 feet 11. 251.7 meters 12. 2202.7 feet 13. 61.1 meters 14. 22.1 meters 15. 10° 16. 1° 17. 23° 18. x=63°, y=27°, z=63°

  2. The Law of Sines Section 14.1 MM4A6 & MM4A7

  3. Essential Question How do I solve problems using the Law of Sines?

  4. Important information: • To find the area of a triangle using the sine function multiply one-half by the product of two sides and the sine of the included angle. • K = ½ (side)(side)(Sin of included angle) • Law of Sines: the ratio of the sine of an angle to its opposite side is the same for all angles in a triangle.

  5. For ABC, the law of sines states the following:

  6. Example 1 • Find the area of ∆RST to the nearest tenth of a square unit.

  7. Example 2 Find the area of ∆JKL to the nearest tenth of a square unit if m<K=59°, m<L=88°, side k=6.7in, and side l=8in.

  8. Example 3 Find the area of ∆XYZ to the nearest tenth of a square unit.

  9. Assignment • Page 891, 10-23 ALL (make the angle in #22 B instead of C) • Using given information, ROUND all angles to the nearest WHOLE NUMBER before you work the problem!!

  10. Do Now • Find the area of ∆ABC, m<A=51°, m<B=62°, side a=7 in., and side c=5.2

  11. Pg. 891, 10-23 all • 14.1 in.² • 30 ft² • 42.9 in.² • 24.7 cm² • 25.2 km² • 368.2 ft² • 97.7 in.² • 160.3 m² 18. 74.2 ft² 19. 19.9 cm² 20. 20.7 km² 21. 1756.2 ft² 22. 13.7m² 23. 722.5 ft²

  12. Example 4 Use the Law of Sines to find a & b.

  13. Example 5 Solve ∆ PQR given the m<Q=115°, m<P=32°, and side p=5.6

  14. Pg. 891, 24-29 • 8.6 • 8.3 • 13.8 • 9.6 • 15.8 • 17.1

  15. Homework Pg. 891, #’s 24-43 all

  16. Do Now • Find the area of ∆ABC, m<A=57°, m<B=68°, side a=5.6 in., and side c=7.1 • Solve ∆ XYZ given the m<Y=119°, m<X=47°, and side x=6.5

  17. Pg. 891, 29-43 ODD 25. c = 8.3 27. a = 9.6 • c = 17.1 31. A=101, b=3.5, a=7.5 33. C=83, b=12.3, c=13.8 35. A=80, b=10, c=3.5 37. A=97, a=20.8, c=15.5 39. B=101, a=12.3, c=13.6

  18. Continued… 41. A=80, a=13.2, b=11.4 43. B=112, a=8.4, c=4.6

  19. Pg. 891, 24-42 even 24. b = 8.6 26. b = 13.8 28. a = 15.8 30. C=85, a=17.3, c=25.3 32. B=100, b=28.8, c=25.3 34. C=95, a=5.9, b=13.3 36. C=80, b=7.4, c=11.4

  20. Continued… 38. A=80, a=12.3, c=10.8 40. C=95, a=10.3, b=11.4 42. A=105, b=4.7, c=6.6

  21. Info… • Quiz tomorrow over 14.1: finding the area of a triangle and solving triangles using the Law of Sines • Today, do page 985, 14.1, 1-14 • Worksheet

  22. Pg. 985, 14.1, 1-14 all • 144.5 in² • 16.6 m² • 94 cm² • 8.7 in² • b=8.7 • c=54.1 • c=7.6 • b=15.6 9. B=95, a=8.2, c=3.1 10. A=123, b=16.3, c=28.7 11. B=55, a=4.4, c=11.8 12. A=98, b=27.1, c=18.7 13. B=86, b=15.6, c=10.9 14. B=45, a=19.2, c=19.9

  23. Workbook pg. 88, 1-12 • 25.5 • 24.8 • 16.7 • 29.7 • 5.6 • 28.9 • C=38, b=4.4, c=3.1 • A=35, c=21.7, a=12.7 9. C=45, a=25.8, c=18.5 10. B=44, a=5.5, b=4.0 11. C=82, a=15.5, c=21.4 12. A=75, a=82.9, c=78.4 16. 1532.7 ft.

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