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Sine Law Ambiguous Case

Sine Law Ambiguous Case
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2015-10-25 10:37
TrigonometryInequalityGeometric Construction
This example demonstrates the ambiguous case of the sine law in SSA triangles, where knowing two side lengths and an opposite angle can lead to two possible solutions. It involves core concepts like the sine law, triangle uniqueness, and inequality analysis. The construction uses commands such as Intersect, Point, Ray, Rotate, Angle, Circle, Segment, OrthogonalLine, and If. Suitable for high school mathematics education to help students understand the applicability and special cases of the sine law.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → F
  3. Ray(A, F) → c_1
  4. Rotate(F, A_1, A) → F'
  5. Angle(F, A, F') → β
  6. Ray(A, F') → d_1
  7. Circle(A, b_1) → e
  8. Intersect(e, d_1, 1.0) → C
  9. Segment(A, C) → b
  10. Circle(C, a_1) → c
  11. Intersect(c, c_1) → D, B
  12. Segment(C, B) → a
  13. Segment(C, D) → g
  14. Segment(A, D) → i
  15. Point(i) → E
  16. Segment(A, B) → j
  17. Point(j) → G
  18. OrthogonalLine(C, c_1) → k
  19. Intersect(c, k) → H, I
  20. Segment(H, C) → a_2
  21. Angle(C, D, A) → γ
  22. Angle(C, B, A) → δ
  23. OrthogonalLine(C, c_1) → d
  24. Intersect(d, c_1) → J
  25. Segment(C, J) → l
  26. Segment(C, B) → p
  27. If(a_1 < b_1, If[l ≟ a_2, "(" + b + ")sin(" + β + ") = " + l + " = " + a_2, If[l > a_2, "(" + b + ")sin(" + β + ") = " + l + " > " + a_2, "(" + b + ")sin(" + β + ") = " + l + " < " + a_2]]) → text3
  28. If(a_1 ≥ b_1, "Yes!", If[b_1 * sin(A_1) > a_1, "No! (We don't even have a triangle!)", If[δ ≟ 90.0°, "No (Hypotenuse-Side Congruence)", "No (Two Possible Triangles)"]]) → text6