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Unit Circle Trig Values for 15° Angles

Unit Circle Trig Values for 15° Angles
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2016-05-04 05:22
CircleAngleTrigonometryParametric EquationGeometric Construction
This GeoGebra interactive lets users touch an angle or point on the unit circle to instantly display its cosine and sine values. The example specifically includes 15° and all its multiples, providing exact radical expressions for each. Core mathematical concepts covered include the unit circle definition, trigonometric function evaluation, exact values of sine and cosine at 15° increments, and the relationship between radian and degree measures. Construction highlights, derived from the GeoGebra command summary, involve the use of Intersect, Circle, Point, Angle, If, Segment, CircleArc, and Numerator/Denominator commands, along with custom RadianConstant objects (e.g., RadianConstant015, RadianConstant330) to generate and display precise angle markings and exact trig values. The interactive is well-suited for high school trigonometry courses, precalculus units on the unit circle, and anyone exploring exact trigonometric values or parametric angle visualization.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → TriangleBaseMeetsHypotenuse
  2. Circle(TriangleBaseMeetsHypotenuse, r) → c
  3. Point(c) → CirclePoint
  4. Angle(Point000, TriangleBaseMeetsHypotenuse, CirclePoint) → θ
  5. If(PiRadians, Text["0π", (1, 0), false, true], Text["0τ", (1, 0), false, true]) → RadianConstant000
  6. If(TauRadians ∧ RadianCommonDenominator, true, false) → RadianTauOver24
  7. If(PiRadians ∧ RadianCommonDenominator, true, false) → RadianPiOver12
  8. If(PiRadians ∧ (¬RadianCommonDenominator), true, false) → RadianPiSimplified
  9. If(RadianPiOver12, Text["\frac{ 23π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 23π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 23τ }{ 24 }", (1, 0), false, true], Text["\frac{ 23τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant345
  10. If(RadianPiOver12, Text["\frac{ 19π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 19π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 19τ }{ 24 }", (1, 0), false, true], Text["\frac{ 19τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant285
  11. If(RadianPiOver12, Text["\frac{ 17π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 17π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 17τ }{ 24 }", (1, 0), false, true], Text["\frac{ 17τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant255
  12. If(RadianPiOver12, Text["\frac{ 7π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 7π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 7τ }{ 24 }", (1, 0), false, true], Text["\frac{ 7τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant105
  13. If(RadianPiOver12, Text["\frac{ 22π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 11π }{ 6 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 22τ }{ 24 }", (1, 0), false, true], Text["\frac{ 11τ }{ 12 }", (1, 0), false, true]]]) → RadianConstant330
  14. If(RadianPiOver12, Text["\frac{ 20π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 5π }{ 3 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 20τ }{ 24 }", (1, 0), false, true], Text["\frac{ 5τ }{ 6 }", (1, 0), false, true]]]) → RadianConstant300
  15. If(RadianPiOver12, Text["\frac{ 16π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 4π }{ 3 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 16τ }{ 24 }", (1, 0), false, true], Text["\frac{ 2τ }{ 3 }", (1, 0), false, true]]]) → RadianConstant240
  16. If(RadianPiOver12, Text["\frac{ 14π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 7π }{ 6 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 14τ }{ 24 }", (1, 0), false, true], Text["\frac{ 7τ }{ 12 }", (1, 0), false, true]]]) → RadianConstant210
  17. If(RadianPiOver12, Text["\frac{ 10π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 5π }{ 6 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 10τ }{ 24 }", (1, 0), false, true], Text["\frac{ 5τ }{ 12 }", (1, 0), false, true]]]) → RadianConstant150
  18. If(RadianPiOver12, Text["\frac{ 8π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 2π }{ 3 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 8τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 3 }", (1, 0), false, true]]]) → RadianConstant120
  19. If(RadianPiOver12, Text["\frac{ 6π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ π }{ 2 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 6τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 4 }", (1, 0), false, true]]]) → RadianConstant090
  20. If(RadianPiOver12, Text["\frac{ 4π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ π }{ 3 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 4τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 6 }", (1, 0), false, true]]]) → RadianConstant060
  21. If(RadianPiOver12, Text["\frac{ 3π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ π }{ 4 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 3τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 8 }", (1, 0), false, true]]]) → RadianConstant045
  22. If(RadianPiOver12, Text["\frac{ 1π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 1τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant015
  23. If(RadianPiOver12, Text["\frac{ 2π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ π }{ 6 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 2τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 12 }", (1, 0), false, true]]]) → RadianConstant030
  24. If(RadianPiOver12, Text["\frac{ 9π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 3π }{ 4 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 9τ }{ 24 }", (1, 0), false, true], Text["\frac{ 3τ }{ 8 }", (1, 0), false, true]]]) → RadianConstant135
  25. If(RadianPiOver12, Text["\frac{ 12π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["π", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 12τ }{ 24 }", (1, 0), false, true], Text["\frac{ τ }{ 2 }", (1, 0), false, true]]]) → RadianConstant180
  26. If(RadianPiOver12, Text["\frac{ 15π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 5π }{ 4 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 15τ }{ 24 }", (1, 0), false, true], Text["\frac{ 5τ }{ 8 }", (1, 0), false, true]]]) → RadianConstant225
  27. If(RadianPiOver12, Text["\frac{ 18π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 3π }{ 2 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 18τ }{ 24 }", (1, 0), false, true], Text["\frac{ 3τ }{ 4 }", (1, 0), false, true]]]) → RadianConstant270
  28. If(RadianPiOver12, Text["\frac{ 21π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 7π }{ 4 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 21τ }{ 24 }", (1, 0), false, true], Text["\frac{ 7τ }{ 8 }", (1, 0), false, true]]]) → RadianConstant315
  29. If(RadianPiOver12, Text["\frac{ 5π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 5π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 5τ }{ 24 }", (1, 0), false, true], Text["\frac{ 5τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant075
  30. If(RadianPiOver12, Text["\frac{ 11π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 11π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 11τ }{ 24 }", (1, 0), false, true], Text["\frac{ 11τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant165
  31. If(RadianPiOver12, Text["\frac{ 13π }{ 12 }", (1, 0), false, true], If[RadianPiSimplified, Text["\frac{ 13π }{ 12 }", (1, 0), false, true], If[RadianTauOver24, Text["\frac{ 13τ }{ 24 }", (1, 0), false, true], Text["\frac{ 13τ }{ 24 }", (1, 0), false, true]]]) → RadianConstant195
  32. If(θ ≟ 0°, RadianConstant000, If[θ ≟ 15°, RadianConstant015, If[θ ≟ 30°, RadianConstant030, If[θ ≟ 45°, RadianConstant045, If[θ ≟ 60°, RadianConstant060, If[θ ≟ 90°, RadianConstant090, If[θ ≟ 120°, RadianConstant120, If[θ ≟ 135°, RadianConstant135, If[θ ≟ 150°, RadianConstant150, If[θ ≟ 180°, RadianConstant180, If[θ ≟ 210°, RadianConstant210, If[θ ≟ 225°, RadianConstant225, If[θ ≟ 240°, RadianConstant240, If[θ ≟ 270°, RadianConstant270, If[θ ≟ 300°, RadianConstant300, If[θ ≟ 315°, RadianConstant315, If[θ ≟ 330°, RadianConstant330, If[θ ≟ 75°, RadianConstant075, If[θ ≟ 105°, RadianConstant105, If[θ ≟ 165°, RadianConstant165, If[θ ≟ 195°, RadianConstant195, If[θ ≟ 255°, RadianConstant255, If[θ ≟ 285°, RadianConstant285, If[θ ≟ 345°, RadianConstant345, ""]]]]]]]]]]]]]]]]]]]]]]]) → RadianConstant
  33. If(ShowRadiusOfOne ∨ ShowRadiusOfR, true, false) → ShowRadiusOfOneOrR
  34. If(θ ≟ 75° ∨ θ ≟ 105°, Text["\sqrt{6}+\sqrt{2}", (1, 0), false, true], If[θ ≟ 15° ∨ θ ≟ 165°, Text["\sqrt{6}-\sqrt{2}", (1, 0), false, true], If[θ ≟ 195° ∨ θ ≟ 345°, Text["\sqrt{2}-\sqrt{6}", (1, 0), false, true], If[θ ≟ 255° ∨ θ ≟ 285°, Text["-\sqrt{6}-\sqrt{2}", (1, 0), false, true], ""]]]) → TriangleHeightTextConstantMultipleOf15r4
  35. If(θ ≟ 15° ∨ θ ≟ 165°, Text["\frac{\sqrt{6}-\sqrt{2}}{4}", (1, 0), false, true], If[θ ≟ 75° ∨ θ ≟ 105°, Text["\frac{\sqrt{6}+\sqrt{2}}{4}", (1, 0), false, true], If[θ ≟ 255° ∨ θ ≟ 285°, Text["-\frac{\sqrt{6}+\sqrt{2}}{4}", (1, 0), false, true], If[θ ≟ 195° ∨ θ ≟ 345°, Text["\frac{\sqrt{2}-\sqrt{6}}{4}", (1, 0), false, true], ""]]]) → TriangleHeightTextConstantMultipleOf15r1
  36. If(Mod[θ, 30°] ≟ 0 ∨ Mod[θ, 45°] ≟ 0, false, true) → ThetaEquals15OddMultiple
  37. If(RadiusOption ≟ 2 ∧ ThetaEquals15OddMultiple, true, false) → ShowRadiusOfFour
  38. If(θ ≟ 15° ∨ θ ≟ 345°, Text["\sqrt{6}+\sqrt{2}", (1, 0), false, true], If[θ ≟ 75° ∨ θ ≟ 285°, Text["\sqrt{6}-\sqrt{2}", (1, 0), false, true], If[θ ≟ 105° ∨ θ ≟ 255°, Text["\sqrt{2}-\sqrt{6}", (1, 0), false, true], If[θ ≟ 165° ∨ θ ≟ 195°, Text["-\sqrt{6}-\sqrt{2}", (1, 0), false, true], ""]]]) → TriangleBaseTextConstantMultipleOf15r4
  39. If(θ ≟ 15° ∨ θ ≟ 345°, Text["\frac{\sqrt{6}+\sqrt{2}}{4}", (1, 0), false, true], If[θ ≟ 75° ∨ θ ≟ 285°, Text["\frac{\sqrt{6}-\sqrt{2}}{4}", (1, 0), false, true], If[θ ≟ 105° ∨ θ ≟ 255°, Text["\frac{\sqrt{2}-\sqrt{6}}{4}", (1, 0), false, true], If[θ ≟ 165° ∨ θ ≟ 195°, Text["-\frac{\sqrt{6}+\sqrt{2}}{4}", (1, 0), false, true], ""]]]) → TriangleBaseTextConstantMultipleOf15r1
  40. If(¬ThetaEquals15OddMultiple, SurdText[cos(θ)], If[ThetaEquals15OddMultiple, TriangleBaseTextConstantMultipleOf15r1, ""]) → CosineGeneralText
  41. Segment(CirclePoint, TriangleBaseMeetsHeight) → TriangleHeight
  42. Segment(TriangleBaseMeetsHypotenuse, TriangleBaseMeetsHeight) → TriangleBase
  43. CircleArc(TriangleBaseMeetsHypotenuse, Point000, CirclePoint) → e
  44. Segment(TriangleBaseMeetsHypotenuse, CirclePoint) → TriangleHypotenuse
  45. Point(c) → Point120
  46. Point(c) → Point135
  47. Intersect(c, xAxis, 1) → Point180
  48. Intersect(c, yAxis, 1) → Point270
  49. Numerator(e / π) → RadianNumerator
  50. Denominator(e / π) → RadianDenominator
  51. Corner(5) → ScreenSizeInPixels
  52. If(ShowRadiusOfR, " r", "") → RadianUnitText
  53. If(x(CirclePoint) < 0, "-", "") → XSignText
  54. If(y(CirclePoint) < 0, "-", "") → YSignText
  55. If(ShowRadiusOfOneOrR ∧ (¬ThetaEquals15OddMultiple), SurdText[sin(θ)], If[ShowRadiusOfTwo ∧ (¬ThetaEquals15OddMultiple), SurdText[(2 * sin(θ))], If[ThetaEquals15OddMultiple ∧ (¬ShowRadiusOfFour), TriangleHeightTextConstantMultipleOf15r1, TriangleHeightTextConstantMultipleOf15r4]]) → TriangleHeightTextConstant
  56. If(ShowRadiusOfOneOrR ∧ (¬ThetaEquals15OddMultiple), SurdText[cos(θ)], If[ShowRadiusOfTwo ∧ (¬ThetaEquals15OddMultiple), SurdText[(2 * cos(θ))], If[ThetaEquals15OddMultiple ∧ (¬ShowRadiusOfFour), TriangleBaseTextConstantMultipleOf15r1, TriangleBaseTextConstantMultipleOf15r4]]) → TriangleBaseTextConstant
  57. If(Degrees, DegreeText, RadianText) → AngleText
  58. If(x(CirclePoint) / y(CirclePoint) < 0, "-", "") → XYSignText
  59. SurdText(tan(θ)) → TangentGeneralText
  60. If(¬ThetaEquals15OddMultiple, SurdText[csc(θ)], If[θ ≟ 75° ∨ θ ≟ 105°, Text["\sqrt{6}+\sqrt{2}", (1, 0), false, true], If[θ ≟ 15° ∨ θ ≟ 165°, Text["\sqrt{6}-\sqrt{2}", (1, 0), false, true], If[θ ≟ 255° ∨ θ ≟ 285°, Text["-\sqrt{6}-\sqrt{2}", (1, 0), false, true], If[θ ≟ 195° ∨ θ ≟ 345°, Text["\sqrt{2}-\sqrt{6}", (1, 0), false, true], ""]]]]) → CosecantGeneralText
  61. If(¬ThetaEquals15OddMultiple, SurdText[sec(θ)], If[θ ≟ 15° ∨ θ ≟ 345°, Text["\sqrt{6}+\sqrt{2}", (1, 0), false, true], If[θ ≟ 75° ∨ θ ≟ 285°, Text["\sqrt{6}-\sqrt{2}", (1, 0), false, true], If[θ ≟ 165° ∨ θ ≟ 195°, Text["-\sqrt{6}-\sqrt{2}", (1, 0), false, true], If[θ ≟ 105° ∨ θ ≟ 255°, Text["\sqrt{2}-\sqrt{6}", (1, 0), false, true], ""]]]]) → SecantGeneralText
  62. SurdText(cot(θ)) → CotangentGeneralText
  63. If(RadiusOption ≟ 1, "1", If[RadiusOption ≟ 2 ∧ (¬ThetaEquals15OddMultiple), "2", If[RadiusOption ≟ 2 ∧ ThetaEquals15OddMultiple, "4", RadiusOption ≟ 3, "r", "r"]]) → TriangleHypText
  64. If(¬ThetaEquals15OddMultiple, SurdText[sin(θ)], If[ThetaEquals15OddMultiple, TriangleHeightTextConstantMultipleOf15r1, ""]) → SineGeneralText
  65. If(TauRadians ∧ (¬RadianCommonDenominator), true, false) → RadianTauSimplified