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Odd and Even Properties of Sine and Cosine

Odd and Even Properties of Sine and Cosine
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2013-04-05 07:35
FunctionTrigonometryGeometric Construction
This example demonstrates the odd and even properties of the sine and cosine functions. By constructing the graphs of these functions, one can observe that the sine function is odd, while the cosine function is even. The construction involves commands such as rotation, mirroring, and angle measurement. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → Origin
  2. Point(Unit) → C
  3. OrthogonalLine(C, xAxis) → a
  4. Intersect(xAxis, a) → D
  5. Segment(C, D) → sine
  6. Segment(Origin, C) → r
  7. CircleSector(Origin, B, C) → θ_1
  8. OrthogonalLine(C, r) → b
  9. Intersect(yAxis, b) → Cosecant
  10. Intersect(xAxis, b) → Secant
  11. Segment(C, Secant) → Tan
  12. Segment(C, Cosecant) → Cot
  13. CircleArc(Origin, B, C) → θ
  14. OrthogonalLine(C, yAxis) → c
  15. Segment(Origin, D) → cos
  16. Rotate(B, 0.5235987755982988, Origin) → B'_1
  17. Rotate(B, 0.7853981633974483, Origin) → B'_2
  18. Rotate(B, 1.0471975511965976, Origin) → B'
  19. Intersect(Unit, yAxis) → A, E
  20. Rotate(E, 0.5235987755982988, Origin) → E'_1
  21. Rotate(E, 0.7853981633974483, Origin) → E'_2
  22. Rotate(E, 1.0471975511965976, Origin) → E'
  23. Intersect(Unit, xAxis) → F, G
  24. Mirror(E'_1, xAxis) → H
  25. Mirror(E'_2, xAxis) → I
  26. Mirror(E', xAxis) → E''
  27. Mirror(B'_1, xAxis) → J
  28. Mirror(B'_2, xAxis) → K
  29. Mirror(B', xAxis) → B''
  30. OrthogonalLine(C, yAxis) → d
  31. Intersect(yAxis, d) → L
  32. Segment(L, C) → e
  33. Segment(L, Origin) → f
  34. TableText({{"csc θ = ", csc(θ)}, {"sec θ = ", sec(θ)}, {"cot θ =", cot(θ)}}) → text4
  35. Angle(G, Origin, C) → α
  36. Mirror(C, xAxis) → C'
  37. Angle(C', Origin, G) → β
  38. Segment(Origin, C') → h
  39. Segment(D, C') → i
  40. TableText({{"sin θ = ", sin(θ)}, {"cos θ =", x(C)}, {"tan θ =", tan(θ)}}) → sin_cos
  41. TableText({{"\sin ( -θ) = ", (-sin(θ))}, {"\cos (-θ) =", x(C)}, {"\tan ( -θ) =", (-tan(θ))}}) → text1

AI Drawing Prompt Reference

Draw the graphs of sine and cosine functions to demonstrate their odd and even properties. Start by plotting the coordinate axes and the origin, then draw a unit circle with an angle θ. Plot the sine and cosine waves in different colors. Rotate the angle θ to various positions and reflect it to show that sine is an odd function (symmetric about the origin) and cosine is an even function (symmetric about the y-axis). Label the values of sine, cosine, tangent, cotangent, secant, and cosecant in different colors.
The prompt above was compiled from the original drawing commands and is for reference only.