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CAST Rule Introduction in Trigonometry

CAST Rule Introduction in Trigonometry
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2012-12-05 07:56
CircleAngleFunctionTrigonometryGeometric Construction
This example demonstrates the application of the CAST rule in trigonometry using the unit circle, visually illustrating the sign changes of sine, cosine, and tangent in different quadrants. Key math concepts include trigonometric functions, unit circle, quadrant division, and the CAST rule. GeoGebra commands and objects used in construction are Circle, Point, Line, Polygon, Angle, Midpoint, and Mirror to build the unit circle and quadrant indicators. Suitable for high school students learning trigonometric functions as an auxiliary teaching tool to deepen understanding of the CAST rule and its application.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 5) → d
  3. Point(d) → B
  4. Line(B, yAxis) → w
  5. Intersect(w, xAxis) → C
  6. Polygon(O, C, B) → poly, b, a, c
  7. Angle(B, C, O) → α
  8. Angle(C, O, B) → β
  9. Midpoint(C, O) → D
  10. Midpoint(B, O) → E
  11. Midpoint(C, B) → F
  12. Point(xAxis) → P
  13. Angle(P, O, B) → γ
  14. Mirror(O, O) → O'
  15. Mirror(C, O) → C'
  16. Mirror(B, O) → B'_1
  17. Polygon(O', C', B'_1) → poly'_1, b'_1, a'_1, c'_1
  18. Mirror(O, x = 0) → O'_1
  19. Mirror(C, x = 0) → C'_1
  20. Mirror(B, x = 0) → B'
  21. Polygon(O'_1, C'_1, B') → poly', b', a', c'
  22. Mirror(O, y = 0) → O'_2
  23. Mirror(C, y = 0) → C'_2
  24. Mirror(B, y = 0) → B'_2
  25. Polygon(O'_2, C'_2, B'_2) → poly'_2, b'_2, a'_2, c'_2
  26. Mirror(E, x = 0) → E'
  27. Mirror(E, y = 0) → E'_2
  28. Mirror(E, O) → E'_1
  29. Mirror(D, O) → D'
  30. Mirror(F, O) → F'_1
  31. Mirror(F, x = 0) → F'
  32. Mirror(F, y = 0) → F'_2
  33. Angle(P, O, B') → γ'
  34. Angle(P, O, B'_1) → γ'_1
  35. Angle(P, O, B'_2) → γ'_2
  36. Angle(B', C', O') → α'
  37. Angle(B'_1, C'_1, O'_1) → α'_1
  38. Angle(B'_2, C'_2, O'_2) → α'_2
  39. Angle(C', O', B') → β'
  40. Angle(C'_1, O'_1, B'_1) → β'_1
  41. Angle(C'_2, O'_2, B'_2) → β'_2
  42. Midpoint(D', O) → G
  43. Midpoint(E'_1, O) → H
  44. Midpoint(G, H) → I
  45. Angle(P, O, B) → δ
  46. Midpoint(D, O) → J
  47. Midpoint(E, O) → K
  48. Midpoint(K, J) → L

AI Drawing Prompt Reference

Draw a unit circle with the intersection points of the x-axis and y-axis marked as the origin. In the first quadrant, draw a triangle connecting the origin to a point on the x-axis, then connecting that point to a point on the y-axis, and finally connecting that point back to the origin. Label the angles of the triangle in different colors. Show the sign changes of sine, cosine, and tangent in different quadrants to illustrate the CAST rule.
The prompt above was compiled from the original drawing commands and is for reference only.