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Cosine and Sine (45-45-90 Triangle)

Cosine and Sine (45-45-90 Triangle)
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2016-04-23 20:44
TriangleFunctionTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the calculation and visualization of cosine and sine function values in a 45-45-90 special right triangle. It involves core mathematical concepts such as the Law of Cosines, Law of Sines, properties of special triangles, and the definition of trigonometric functions. The construction utilizes GeoGebra commands like Intersect, Segment, Point, Rotate, Angle, Line, OrthogonalLine, Polygon, and Mirror. It is suitable for high school students learning the geometric properties of special triangles and their relationship with trigonometric functions.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → B
  2. Segment(B, A) → a
  3. Point(Circle[C, 1]) → D
  4. Segment(C, D) → b
  5. Rotate(C, 45°, D) → C'
  6. Angle(C, D, C') → α
  7. Line(D, C') → d
  8. OrthogonalLine(C, d) → e
  9. Intersect(d, e) → E
  10. Polygon(D, C, E) → poly1, e_1, d_1, c_1
  11. Angle(poly1) → β, γ, δ
  12. Point(Circle[F, Segment[D, C]]) → G
  13. Rotate(G + (Segment[C, E], 0), Angle[E - C] + Angle[G - F] - Angle[C - D], G) → H
  14. Polygon(F, G, H) → poly2, h, f, g
  15. Point(Circle[I, Segment[D, C]]) → J
  16. Rotate(J + (Segment[C, E], 0), Angle[E - C] + Angle[J - I] - Angle[C - D], J) → K
  17. Polygon(I, J, K) → poly3, k, i, j
  18. Angle(poly2) → ε, ζ, η
  19. Point(Circle[L, Segment[D, C]]) → M
  20. Rotate(M + (Segment[C, E], 0), Angle[E - C] + Angle[M - L] - Angle[C - D], M) → N
  21. Polygon(L, M, N) → poly4, n, l, m
  22. Mirror(L, n) → L'
  23. Mirror(M, n) → M'
  24. Mirror(N, n) → N'
  25. Polygon(L', M', N') → poly4', n', l', m'
  26. Point(Circle[O, Segment[D, C]]) → P
  27. Rotate(P + (Segment[C, E], 0), Angle[E - C] + Angle[P - O] - Angle[C - D], P) → Q
  28. Polygon(O, P, Q) → poly5, q, o, p