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Sine Formula for Sum of Two Angles by zhang hongwei

Sine Formula for Sum of Two Angles by zhang hongwei
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2019-06-05 17:06
AngleTrigonometryGeometric Construction
In collections My First GGB Storybook
This example demonstrates the derivation of the sine formula for the sum of two angles using geometric figures and trigonometric functions. Key mathematical concepts include trigonometry, angle relationships, and geometric construction. Construction involves GeoGebra commands such as Intersect, Circle, Point, Segment, Line, Angle. Suitable for middle or high school mathematics to understand trigonometric properties and geometric applications.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 1 / 2) → c
  3. Point(c) → A
  4. Intersect(c, Line[O, A], 2) → B
  5. Segment(A, B) → g
  6. Point(c) → C
  7. Segment(A, C) → h
  8. Segment(B, C) → i
  9. Line(C, y = 0) → f
  10. Intersect(OrthogonalLine[A, f], f) → D
  11. Intersect(OrthogonalLine[B, f], f) → E
  12. Intersect(Line[B, f], OrthogonalLine[A, f]) → F
  13. Segment(B, E) → j
  14. Segment(B, F) → k
  15. Segment(D, F) → l
  16. Segment(D, E) → m
  17. Angle(D, C, A) → α
  18. Angle(C, B, A) → β
  19. Angle(E, B, C) → γ
  20. Angle(F, A, B) → δ
  21. Angle(A, C, B) → ε
  22. Angle(B, F, D) → ζ
  23. Angle(F, D, E) → η
  24. Angle(D, E, B) → θ
  25. Text("1", Midpoint[A, B] + (0, -0.05)) → text1
  26. Text(RotateText["\purple{\cos(α+β)}", 90°], F - (1 / 4 * (F - A)), true, true) → text2
  27. Text("\purple{\sin(α+β)}", Midpoint[B, F] + (-0.08, 0.05), true, true) → text3
  28. Text("$\red{\sin(α)}\blue{\cos(β)}$", E + (0.02, 0), true, true) → text4
  29. Text("\red{\cos(α)}\blue{\sin(β)}", Midpoint[C, D] + (-0.06, 0), true, true) → text5

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with radius one. Place three points on the circle, with two of them forming a diameter through the center, and the third point anywhere on the circle. Connect these points to form a triangle. From two vertices of the triangle, drop perpendiculars to the horizontal diameter, and extend one perpendicular to intersect the other, creating several right triangles. Label the angles, including the sum of two angles, their difference, and the triangle's interior angles. Add text annotations in different colors to label the cosine and sine product terms, and mark the unit length.
The prompt above was compiled from the original drawing commands and is for reference only.