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Deriving the Tangent Graph from the Unit Circle

Deriving the Tangent Graph from the Unit Circle
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2018-03-02 19:06
CircleAngleFunctionTrigonometryGraphing
This GeoGebra example dynamically demonstrates how the tangent function (tan θ) graph is derived from the unit circle through geometric construction. It visualizes the relationship between an angle θ rotating on the unit circle and the corresponding tangent value plotted on a coordinate plane. Core mathematical concepts include the unit circle, definitions of sine, cosine, and tangent, angle measurement in radians, and the generation of a function graph. The construction uses GeoGebra commands such as Point, Rotate, Segment, Angle, Circle, Vector, Sequence, and Function to create the circle, rotating radius, angle labels, and a sequence of points that trace the tangent curve. This resource is ideal for high school or introductory college students learning trigonometry, as well as for teachers demonstrating the periodic nature, asymptotes, and geometric origin of the tangent function.

GeoGebra Commands & Parameters

  1. Point(Ray[(0, 0), (-1, 0)]) → B
  2. Rotate(A, θ, B) → P
  3. Segment(A, Intersect[Line[B, P], x = x(A)]) → h
  4. Segment(P, Intersect[Line[B, P], x = x(A)]) → g
  5. Rotate(A, θ, B) → A'
  6. Angle(A, B, A') → α
  7. Rotate(A, θ, B) → A'_1
  8. Angle(A, B, A'_1) → β
  9. Segment(Q, (x(Q), 0)) → e
  10. Circle(B, A) → c
  11. Segment(B, P) → a
  12. Vector((x(B), y(Corner[1])), (x(B), y(Corner[4]))) → u
  13. Vector((0, y(Corner[1])), (0, y(Corner[4]))) → w
  14. Vector((0, 0), (x(Corner[2]), 0)) → z
  15. Sequence(Segment[((0.75 * i), 0), ((0.75 * i), -0.025)], i, 1, 4) → list1
  16. Vector(B - (1.2, 0), B + (1.2, 0)) → v
  17. Sequence(Segment[(0, (0.5 * i) - 1), (-0.025, (0.5 * i) - 1)], i, 0, 4) → list2
  18. Sequence(Segment[(x(B), (0.5 * i) - 1), (x(B) - 0.025, (0.5 * i) - 1)], i, 0, 4) → list3
  19. Sequence(Segment[B + ((0.5 * i) - 1, 0), B + ((0.5 * i) - 1, -0.025)], i, 0, 4) → list4
  20. Segment(C, Q) → d
  21. Function(tan((120° * x)), 0, (θ / ((360 * γ)) * 3)) → f
  22. Point(Ray[(0, 0), (-1, 0)]) → B_1
  23. Rotate(A_1, θ, B_1) → P_1
  24. Rotate(A_1, θ, B_1) → A'_2
  25. Rotate(A_1, θ, B_1) → A'_3
  26. Circle(B_1, A_1) → c_1