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Unit Circle and Trigonometric Functions by Antalcides Olivo

Unit Circle and Trigonometric Functions by Antalcides Olivo
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2018-09-18 11:38
CircleFunctionTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates how to trace the graphs of circular trigonometric functions from a unit circle centered at the origin. It involves core concepts like the unit circle, trigonometric functions (sine, cosine), vectors, and tangents. Construction uses GeoGebra commands such as Circle, OrthogonalLine, Intersect, Polygon, Segment, Line, Tangent, and CircleArc. Suitable for high school mathematics education to understand the geometric origin of trigonometric function graphs.

GeoGebra Commands & Parameters

  1. Circle((α, 0), 1) → c
  2. OrthogonalLine(sine_{positive period}, xAxis) → d
  3. Intersect(d, xAxis) → cosinus_{postive period}
  4. Polygon(Center, sine_{positive period}, cosinus_{postive period}) → polígono1, d_1, center, a_1
  5. OrthogonalLine(sine_{positive period}, yAxis) → e
  6. Intersect(e, yAxis) → C
  7. Segment(sine_{positive period}, C) → f
  8. Segment(sine, C) → g
  9. Intersect(c, xAxis, 2) → Intersection_{tangent}
  10. Line(Center, sine_{positive period}) → generatrix
  11. Tangent(Intersection_{tangent}, c) → h,
  12. Intersect(c, i) → Intersection_{cotangent},
  13. Intersect(i, generatrix) → D
  14. CircleArc(Center, Intersection_{tangent}, sine_{positive period}) → k
  15. Intersect(xAxis, yAxis) → A
  16. Intersect(generatrix, h) → tangent_{positive period}
  17. Intersect(i, generatrix) → cotangent_{positive period}
  18. Intersect(generatrix, h) → secant_{positive period}
  19. Polygon(Center, secant_{positive period}, Intersection_{tangent}) → polígono2, intersection_{tangent}, center_1, b_1
  20. Intersect(i, generatrix) → cosecant_{positive period}
  21. Segment(Center, cosecant_{positive period}) → j
  22. Polygon(cotangent_{positive period}, Intersection_{cotangent}, Center) → polígono3, center_2, cotangent_1, intersection_{cotangent}
  23. Segment(cotangent_{positive period}, Intersection_{cotangent}) → p

AI Drawing Prompt Reference

Draw a unit circle centered at the origin and show how to derive the graphs of sine and cosine functions from the unit circle. First, draw the unit circle, then find the image of the sine function on the positive half-axis and draw the intersection point of this image with the x-axis. Next, connect the origin to the sine function image and draw the tangent line to this image. Then, find the image of the cosine function on the positive half-axis and draw the intersection point of this image with the x-axis. Finally, connect the origin to the cosine function image and draw the tangent line to this image.
The prompt above was compiled from the original drawing commands and is for reference only.