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Determining Exact Values of Sine and Cosine Ratios

Determining Exact Values of Sine and Cosine Ratios
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2018-01-19 00:14
CircleAngleVectorTrigonometryGeometric Construction
This example demonstrates how to geometrically construct and determine the exact values of sine and cosine ratios for specific angles. Key math concepts include trigonometric functions, circles, vectors, and angle measurement. GeoGebra commands and objects used in the construction are Intersect, Point, Circle, Segment, Distance, Angle, Vector, Polygon, and Line. Suitable for high school students learning trigonometry and analytical geometry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Point(xAxis) → E
  3. Circle(O, E) → c_1
  4. Point(c_1) → P
  5. Segment(O, P) → a
  6. Distance(P, A_1) → b
  7. Angle(E, O, P) → α
  8. Segment(A_1, (x(A_1), 0)) → f
  9. Segment(A_2, (x(A_2), 0)) → g
  10. Segment(B_2, B_1) → h
  11. Segment(C_2, C_1) → i
  12. Vector(O, P) → u
  13. Vector(P, (x(P), 0)) → v
  14. Vector(P, (0, y(P))) → w
  15. Point(u) → A
  16. Segment(A, C_2) → e
  17. Segment(A, B_2) → j
  18. Segment(A, A_2) → k
  19. Segment(A, A_3) → l
  20. Segment(A, B_3) → m
  21. Segment(A, C_3) → n
  22. Segment(A, C_4) → p
  23. Segment(A, B_4) → q
  24. Segment(A, A_4) → r
  25. Segment(A, A_1) → s_1
  26. Segment(A, B_1) → t
  27. Segment(A, C_1) → a_1
  28. Polygon(P, (x(P), 0), A) → poly1, a_2, B, b_1
  29. Polygon((abs(x(P)), abs(y(P))), A, (abs(x(P)), 0)) → poly2, d_1, e_1, a_3
  30. Point(d_1) → C
  31. Line(C, C_1) → s
  32. Point(e) → D
  33. Line(C, D) → c
  34. Point(j) → F
  35. Line(C, F) → f_1
  36. Point(k) → G
  37. Line(C, G) → g_1
  38. Point(l) → H
  39. Line(C, H) → h_1
  40. Point(m) → I
  41. Line(C, I) → i_1
  42. Intersect(c_1, c) → J, K
  43. Intersect(c_1, f_1) → L, M
  44. Intersect(c_1, g_1) → P_1, Q
  45. Intersect(c_1, h_1) → R, T
  46. Intersect(c_1, i_1) → U, V
  47. Intersect(c_1, s) → Z, D_1
  48. Segment(Q, P_1) → j_1
  49. Segment(M, L) → l_1
  50. Segment(V, U) → m_1
  51. Segment(D_1, Z) → n_1
  52. Segment(K, J) → p_1
  53. Segment(T, R) → k_1