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Proof of Sine Addition Formula: A Geometric Approach

Proof of Sine Addition Formula: A Geometric Approach
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2018-02-17 19:56
AngleFunctionTrigonometry
This example demonstrates the sine addition formula through geometric figures. Key mathematical concepts include the sine addition formula in trigonometry. Construction highlights the use of circles, angles, and polygons with GeoGebra commands. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → c
  3. Intersect(c, xAxis, 1) → E
  4. Intersect(c, xAxis, 2) → H
  5. Intersect(c, yAxis, 2) → B
  6. CircleArc(A, B, E) → d
  7. CircleArc(A, H, B) → e
  8. Point(d) → C
  9. Point(e) → D
  10. Circle(C, Radius[d]) → f
  11. Circle(D, Radius[e]) → g
  12. Intersect(f, g, 1) → F
  13. Polygon(A, C, F, D) → q1, a, c_1, f_1, d_1
  14. Polygon(C, I, A) → t1, a_1, c_2, i
  15. Polygon(C, G, F) → t2, f_2, c_3, g_1
  16. Polygon(A, I, G, F, D) → poly1, a_2, i_1, g_2, f_3, d_2
  17. If(0 < b < 1, Polygon[Translate[C, Vector[C, Dilate[C, 1 - b, F]]], Translate[I, Vector[C, Dilate[C, 1 - b, F]]], Translate[A, Vector[C, Dilate[C, 1 - b, F]]]], Polygon[Translate[C, Vector[C, Dilate[C, 1 - 1, F]]], Translate[I, Vector[C, Dilate[C, 1 - 1, F]]], Translate[A, Vector[C, Dilate[C, 1 - 1, F]]]]) → t3
  18. If(2 < b < 3, Polygon[Translate[C, Vector[C, Dilate[C, 3 - b, A]]], Translate[G, Vector[C, Dilate[C, 3 - b, A]]], Translate[F, Vector[C, Dilate[C, 3 - b, A]]]], Polygon[Translate[C, Vector[C, Dilate[C, 1 - 1, A]]], Translate[G, Vector[C, Dilate[C, 1 - 1, A]]], Translate[F, Vector[C, Dilate[C, 1 - 1, A]]]]) → t4
  19. Angle(t4) → α, β_1, γ
  20. Angle(t3) → β, ε, ζ
  21. Angle(A, C, F) → δ
  22. Angle(t2) → η, θ, ι
  23. If(4 < b < 5, Segment[G, Dilate[G, 5 - b, J]], Segment[G, Dilate[G, 5 - 5, J]]) → h