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Cos(A-B) and Its Applications

Cos(A-B) and Its Applications
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2020-12-17 04:38
AngleTrigonometryGeometric Construction
This example illustrates the geometric meaning of the cosine(A-B) formula. By constructing points A and B on the unit circle and their angle relationship, the value of cos(A-B) is intuitively demonstrated. Key mathematical concepts include trigonometric functions, the unit circle, and angles. GeoGebra commands used include Circle, Point, Rotate, Polygon, Segment, Angle, Midpoint, CircleArc, and Intersect. Suitable for high school mathematics teaching and learning, particularly in trigonometry and geometric construction.

GeoGebra Commands & Parameters

  1. Circle(O, 1) → UC
  2. Point(UC) → A
  3. Point(UC) → B
  4. Rotate((1, 0), Angle[B, O, A]) → AminB
  5. Polygon(LL, LR, UR, UL) → theBox, ll, lr, ur, ul
  6. Segment(O, A) → Aseg
  7. Segment(O, B) → Bseg
  8. Segment(O, AminB) → ABseg
  9. Angle(P0, O, A) → angA
  10. Midpoint(ABseg) → AminBmid
  11. Midpoint(Aseg) → Asegmid
  12. CircleArc(O, AminBmid, Asegmid) → Bminarc
  13. Intersect(Bminarc, AngularBisector[Aseg, ABseg]) → Bminmid,
  14. Midpoint(Bseg) → Bsegmid
  15. Angle(AminB, O, A) → Bneg
  16. Angle(B, O, A) → AminB1
  17. Angle((1, 0), O, AminB) → AminBs
  18. Segment(A, B) → cordAB
  19. Segment((1, 0), AminB) → cordAminB
  20. Angle((1, 0), O, B) → angB
  21. Textfield(stuAx) → inputAx
  22. Textfield(stuAy) → inputAy
  23. If(step > 2, "B_T&=(\cos B, \sin B)", "") → Bttext
  24. If(step > 1, "A_T&=(\cos A, \sin A)", "") → Attext
  25. If(step > 3, "(A-B)_T&=\Big(\cos(A-B), \sin(A-B)\Big)", "") → ABttext
  26. Textfield(stuBx) → inputBx
  27. Textfield(stuBy) → inputBy
  28. Textfield(stuABy) → inputABy
  29. Textfield(stuABx) → inputABx
  30. Textfield(stud1) → inputd1
  31. Textfield(stud2) → inputd2
  32. If(step > 4, "d_1&=\sqrt{(\cos B-\cos A)^2+(\sin B-\sin A)^2}", "") → d1text
  33. If(step > 4, "d_2&=\sqrt{\big(\cos (A- B)-1\big)^2+ \sin^2( A-B)}", "") → d2text
  34. If(showjust, stepneg1supp, "") → stepneg12

AI Drawing Prompt Reference

Draw a unit circle centered at the origin. Place two movable points A and B on the circle, and draw radii from the center to each. Rotate the point at angle zero by the angle from B to A to create a new point, and draw a radius to it. Construct an arc between the midpoints of this new radius and the radius to A. Connect A and B with a chord, and also connect the point at angle zero to the new point with another chord. Show angle labels and gradually display the coordinates of A, B, and the new point, along with distance formulas for the two chords, to illustrate the cosine difference identity.
The prompt above was compiled from the original drawing commands and is for reference only.