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Trigonometric Curves and the Unit Circle: Dynamic Visualization

Trigonometric Curves and the Unit Circle: Dynamic Visualization
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2016-11-15 05:29
CircleTrigonometrySlider AnimationLocusGraphing
① This GeoGebra example dynamically illustrates the geometric connection between trigonometric curves (sine, secant, etc.) and points on the unit circle, showing how function values evolve as the angle changes. ② Core mathematical concepts include the unit circle definition, sine function, secant line, angle-radian correspondence, and locus generation of function graphs. ③ Construction highlights: uses Intersect to find intersection points, Circle to build the unit circle, Point for moving points, Angle to measure angles, Locus to generate curves, and LaTeX for formulas; objects include center A, radius r, moving point B, angle α, locus loc1, with buttons (button1/2/3) controlling animation and display. ④ Suitable for high school and pre-university students learning trigonometry and coordinate geometry, as well as for teachers demonstrating the dynamic mapping between function graphs and the unit circle.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → C_2
  3. Point(C_2) → B
  4. Intersect(C_2, xAxis, 2) → D
  5. Angle(D, A, B) → α
  6. Locus(S, B) → loc1
  7. LaTeX(loc1, true, true) → text1
  8. Locus(C, B) → loc2
  9. Locus(T, B) → loc3

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with radius 1. Place a movable point on the circle, and draw a radius from the center to this point, marking the angle between this radius and the positive x-axis. As the point moves around the circle, dynamically generate three curves: a sine curve showing the y-coordinate of the point versus the angle, a secant curve showing the x-coordinate of the intersection of the extended radius with the x-axis, and a tangent curve showing the y-coordinate of the intersection of the extended radius with the y-axis. All curves update in sync with the circle, illustrating the geometric relationship between trigonometric functions and points on the unit circle.
The prompt above was compiled from the original drawing commands and is for reference only.