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Unit Circle 7B: Sine, Cosine and Tangent Lines

Unit Circle 7B: Sine, Cosine and Tangent Lines
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2017-11-24 01:34
CircleAngleTrigonometryDerivativeSlider Animation
This GeoGebra example dynamically illustrates the geometric meaning of sine and cosine functions on the unit circle, along with tangent lines and perpendicular constructions. It demonstrates how the coordinates of a point on the unit circle relate to trigonometric ratios, and how the slope of a tangent line connects to the derivative concept. The construction uses commands such as Circle, Point, OrthogonalLine, Angle, CircleArc, Rotate, Line, Segment, and Intersect to build the interactive diagram. Suitable for high school or introductory college students learning trigonometry, unit circle definitions, and basic calculus ideas. It serves as a visual aid for teachers and an exploratory tool for students.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → C
  2. Circle(C, 1) → c_1
  3. Point(c_1) → D
  4. OrthogonalLine(D, xAxis) → g
  5. Intersect(g, xAxis) → A
  6. Angle(A, C, D) → α
  7. Intersect(c_1, xAxis, 2) → D_1
  8. Intersect(c_1, xAxis, 2) → E
  9. Intersect(c_1, yAxis, 2) → F
  10. CircleArc(C, E, F) → d
  11. Rotate(A, β, C) → A'
  12. Angle(A, C, A') → γ
  13. Line(C, A') → f
  14. Intersect(c_1, f, 2) → B
  15. Segment(C, B) → r
  16. OrthogonalLine(B, xAxis) → i
  17. Intersect(i, xAxis) → G
  18. Segment(B, G) → sine
  19. Segment(G, C) → cosine

AI Drawing Prompt Reference

Draw a unit circle centered at the origin on a coordinate plane. Place a point on the circle and drop a perpendicular from it to the horizontal axis, forming a right triangle with the origin. Mark the angle at the origin with an arc, and show the horizontal and vertical projection segments representing cosine and sine. Then choose another point on the circle, draw a line through the origin and this point, and mark the second intersection with the circle. From that second intersection, drop a perpendicular to the horizontal axis, forming another right triangle. Also draw a quarter-circle arc from the positive horizontal axis to the positive vertical axis, and show an angle created by rotation with its arc. The diagram should clearly illustrate the relationship between trigonometric functions and geometric segments on the unit circle.
The prompt above was compiled from the original drawing commands and is for reference only.