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Meaning of Sine, Cosine, Tangent in a Right Triangle

Meaning of Sine, Cosine, Tangent in a Right Triangle
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2017-05-24 00:05
TriangleTrigonometryGeometric Construction
In collections Similarity & Right Triangle Trigonometry Similarity, Right Triangles, and Trigonometry (SRT) Cluster
This example demonstrates the geometric meanings of sine, cosine, and tangent functions within a right triangle context. It involves core math concepts like trigonometric definitions, right triangle side-angle relationships, and geometric transformations. Construction highlights GeoGebra commands such as Point, Segment, OrthogonalLine, Rotate, Angle, Ray, Circle, Intersect, Polygon, Textfield, Midpoint, RotateText, and If. Suitable for high school students learning trigonometric basics and geometric transformations.

GeoGebra Commands & Parameters

  1. Point(Ray[A, A + (1, 0)]) → B
  2. Segment(A, B) → f
  3. OrthogonalLine(B, f) → g
  4. Rotate(B, (a * pi / 180), A) → B'
  5. Angle(B, A, B') → α
  6. Ray(A, B') → h
  7. Circle(B, 2) → c
  8. Intersect(c, g, 2) → C
  9. Ray(B, C) → i
  10. Intersect(h, i) → D
  11. Segment(B, D) → j
  12. Segment(A, D) → k
  13. Polygon(A, B, D) → poly1, d, a_1, b
  14. Rotate(B, If[0 < e ≤ 1, (-((e * (pi / 2 - α)))), α - pi / 2], D) → B'_1
  15. Rotate(D, If[0 < e ≤ 1, (-((e * (pi / 2 - α)))), α - pi / 2], D) → D'
  16. Segment(B'_1, D') → j'
  17. Rotate(A, If[0 < l ≤ 1, (α * l), α], A) → A'
  18. Rotate(B, If[0 < l ≤ 1, (α * l), α], A) → B'_2
  19. Segment(A', B'_2) → f'
  20. Rotate(B, If[0 < m ≤ 1, (m * pi / 2), pi / 2], B) → B'_3
  21. Rotate(D, If[0 < m ≤ 1, (m * pi / 2), pi / 2], B) → D'_1
  22. Segment(B'_3, D'_1) → j'_1
  23. Textfield(a) → InputBox1
  24. Angle(D, B, A) → β
  25. Midpoint(A, D) → E
  26. RotateText(text7, α) → text8
  27. Midpoint(A', B'_2) → F
  28. RotateText(text9, α) → text10
  29. Point(Ray[G, G + (1, 0)]) → H
  30. Segment(G, H) → t
  31. Rotate(H, (a * pi / 180), G) → H'
  32. Ray(G, H') → f_1
  33. OrthogonalLine(H, t) → g_1
  34. Intersect(g_1, f_1) → I
  35. Polygon(G, H, I) → t1, i_1, g_2, h_1
  36. Angle(H, G, I) → γ
  37. Segment(H, I) → j_1
  38. Rotate(H, If[0 < e ≤ 1, ((-e) * (pi / 2 - α)), (-(pi / 2 - α))], I) → H'_{1}
  39. Rotate(I, If[0 < e ≤ 1, ((-e) * (pi / 2 - α)), (-(pi / 2 - α))], I) → I'
  40. Segment(H'_{1}, I') → k_1
  41. Rotate(G, If[0 < l ≤ 1, (l * α), α], G) → G'
  42. Rotate(H, If[0 < l ≤ 1, (l * α), α], G) → H'_{2}
  43. Segment(G', H'_{2}) → t'
  44. Angle(I, H, G) → δ
  45. Rotate(H, If[0 < m ≤ 1, (m * pi / 2), pi / 2], H) → H'_{3}
  46. Rotate(I, If[0 < m ≤ 1, (m * pi / 2), pi / 2], H) → I'_{1}
  47. Segment(H'_{3}, I'_{1}) → l_1
  48. RotateText(text2, atand(Slope[Line[D, A]])) → text16
  49. Textfield(a) → InputBox2
  50. If(e ≠ 0 ∧ (α ≟ pi / 6 ∨ α ≟ pi / 2 ∨ α ≟ 0), "=", "≈") → text6
  51. If(l ≠ 0 ∧ (α ≟ pi / 3 ∨ α ≟ pi / 2 ∨ α ≟ 0), "=", "≈") → text19
  52. If(m ≠ 0 ∧ (α ≟ pi / 4 ∨ α ≟ 0), "=", "≈") → text20