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Sine and Cosine on the Unit Circle (Degrees)

Sine and Cosine on the Unit Circle (Degrees)
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2015-09-27 10:07
CircleAngleTrigonometryGeometric ConstructionSlider Animation
This GeoGebra example visually demonstrates how sine and cosine values are derived from the unit circle as an angle varies in degrees. It highlights the geometric definition of trigonometric functions, showing the relationship between the angle, the point on the circle, and the horizontal/vertical projections. Core concepts include the unit circle, sine, cosine, angle measurement, similarity transformations, and secant lines. The construction employs commands such as Circle, Vector, Angle, OrthogonalLine, Intersect, Polygon, and Segment, involving objects like A, B, C, D, E, c, F, G, H, I, u, v, w, z, a, J, b, d, α, e, K, poly1, a_1, j, k, f, L, g, M, N, h, p, O, P, i, Q, l. It is ideal for high school trigonometry classes, college precalculus review, and interactive learning of periodic functions and their geometric origins.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Vector(A, F) → u
  3. Vector(A, G) → v
  4. Vector(A, H) → w
  5. Vector(A, I) → z
  6. Angle(B, A, J) → α
  7. OrthogonalLine(J, b) → e
  8. Intersect(e, b) → K
  9. Polygon(J, K, A) → poly1, a_1, j, k
  10. OrthogonalLine(J, d) → f
  11. Intersect(f, d) → L
  12. Circle(L, 1.3) → g
  13. Intersect(g, f, 1) → M
  14. Intersect(g, f, 2) → N
  15. Segment(M, N) → h
  16. Circle(K, 1.3) → p
  17. Intersect(p, e, 2) → O
  18. Intersect(p, e, 1) → P
  19. Segment(O, P) → i

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with radius 1. From the center, draw three vectors pointing to different points on the circle. Mark a moving point on the circle and indicate the angle from a fixed starting point to this point using an arc. Drop a perpendicular from this moving point to a horizontal line, marking the foot, and form a right triangle filled with semi-transparent color. Similarly, drop a perpendicular to a vertical line, marking another foot. Around this second foot, draw a small circle that intersects the vertical line at two points, and connect these points with a segment. Around the first foot, draw another small circle intersecting its perpendicular line at two points, and connect those with a segment. The construction visually shows how the horizontal and vertical projections of the point on the circle correspond to cosine and sine as the angle changes.
The prompt above was compiled from the original drawing commands and is for reference only.