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Mastering the Law of Sines: Triangle Solutions & Applications

Explore deriving the Law of Sines, solving triangles with AAS and ASA, the Ambiguous Case with SSA, and practical applications of the Law of Sines in geometry. Includes area calculations and problem-solving examples.

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Mastering the Law of Sines: Triangle Solutions & Applications

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  1. 5.5 Demana, Waits, Foley, Kennedy The Law of Sines

  2. What you’ll learn about • Deriving the Law of Sines • Solving Triangles (AAS, ASA) • The Ambiguous Case (SSA) • Applications … and why The Law of Sines is a powerful extension of the triangle congruence theorems of Euclidean geometry.

  3. Review

  4. Overview • A triangle can be defined as long as we have three of the six parts and one of the parts is a side. • 1) ASA or SAA • 2) SSA (two sides and an angle opposite) • 3) SAS (two sides and an included angle) • 4) SSS • Situations 1 &2 can be solved using Law of Sines • Situations 3 &4 can be solved using Law of Cosines

  5. Area of any triangle • The area of a triangle with sides of lengths a and b and with an included angle θ is; A = 1/2 ab sinθ 10 3

  6. Create your own • Create as many problems as possible from the figure shown:

  7. Law of Sines ***Angles and sides opposite use same letter***

  8. Use area to prove Law of Sines

  9. Example: Solving a Triangle Given Two Angles and a Side

  10. Solution

  11. Solution

  12. A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angeles, 340 miles apart. At an instant when the satellite is between these two stations, its angle of elevation is simultaneously observed to be 60 degrees at Phoenix and 75 degrees at Los Angeles. How far is the satellite from Los Angeles?

  13. Example: Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)

  14. Solve the triangle

  15. Solve triangle ABC, where angle A = 42 degrees and sides a & b are 70 mm & 122 mm respectively

  16. Visual of Ambiguous Case

  17. 5.5 HW, Page 439 • Be able to do 1 – 22; 27 - 35

  18. Solution

  19. Solution

  20. Solution

  21. Solution

  22. Solution

  23. Solution

  24. A 65º x B 15ft 15º C Example: Finding the Height of a Pole

  25. A 65º x B 15ft 15º C Solution

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