File:01-Winkel 4,153° 175,847° Animation.gif

Summary[edit]

Description01-Winkel 4,153° 175,847° Animation.gif	Deutsch: Näherungskonstruktion des Winkels 4,153° mit Supplementwinkel 175,847°, Animation English: Proximity construction of the angle 4.153° with supplementary angle 175.847°, animation
Date	24 October 2016
Source	Own work
Author	Petrus3743
Other versions

Konstruktionsprinzip[edit]

• Einheitskreis Radius r = 1, in GeoGebra,
• Näherungskonstruktion mittels des dritten Strahlensatzes, siehe hierzu auch Konstruktion einer Dezimalzahl (The construction of a decimal number); auch allein mit Zirkel und Lineal möglich.
  • Die Grundkonstruktion (Schema) besteht hauptsächlich aus einem horizontalen und fünf vertikalen Zahlenstrahlen.
• Ist der Winkel 45° ≥ α > 0° wird der Sinus von α bzw. β verwendet.
• Ist der Winkel 45° < α ≤ 135° wird der Kosinus von α bzw. β verwendet.
• Ist der Winkel 135° < α < 180° wird der Sinus von α bzw. β verwendet.

Fehler[edit]

Anmerkung: 1 Winkelsekunde = ${\displaystyle \textstyle {\frac {1^{\circ }}{3600}}}$ = 2,7E-4

• Absoluter Winkelfehler F_α ≈ -4,065E-6°
• Fehler der Sehne A₁F₁

A₁F₁ ist die konstruierte Sehne S_1K = 7,24675865282655E-2 [LE]

Sehne S_1SOLL = ${\displaystyle \sin \left({\frac {\alpha }{2}}\right)\cdot 2}$ = 7,24676574450448E-2 [LE]

Absoluter Fehler der Sehne F_S1 = S_1K - S_1SOLL = -7,09...E-8 [LE]

• Bei einem Radius r = 100 km wäre der Fehler an der Sehne ca. -7,1 mm.

• Absoluter Winkelfehler F_β ≈ 4,065E-6°
• Fehler der Sehne A₁G₁

A₁G₁ ist die konstruierte Sehne S_2K = 1,99868668102401 [LE]

Sehne S_2SOLL = ${\displaystyle \sin \left({\frac {\beta }{2}}\right)\cdot 2}$ = 1,99868667845274 [LE]

Absoluter Fehler der Sehne F_S2 = S_2K - S_2SOLL = 2,57...E-9 [LE]

• Bei einem Radius r = 1.000 km wäre der Fehler an der Sehne ca. 2,6 mm.

Construction principle[edit]

• Unit circle radius r = 1, in GeoGebra,
• Proximity construction by means of the third intercept theorem, see also The construction of a decimal number ; also possible with compass and ruler.
  • The basic construction (scheme) consists mainly of one horizontal and five vertical number lines.
• If the angle 45° ≥ α > 0° the sinus of α resp. β will used.
• If the angle 45° < α ≤ 135° the cosine of α resp. β will used.
• If the angle 135° < α < 180° the sinus of α resp. β will used.

Error[edit]

Note: 1 angular seconds = ${\displaystyle \textstyle {\frac {1^{\circ }}{3600}}}$ = 2.7E-4

• Absolute error of the angle F_α ≈ -4.065E-6°
• Error of the chord A₁F₁

A₁F₁ is the constructed chord S_1K = 7.24675865282655E-2 [unit of length]

Chord S_1SHOULD = ${\displaystyle \sin \left({\frac {\alpha }{2}}\right)\cdot 2}$ = 7.24676574450448E-2 [unit of length]

Absolute error of the chord F_S1 = S_1K - S_1SHOULD = -7.09...E-8 [unit of length]

• If the radius r = 100 km would be the error of the chord approx. -7.1 mm.

• Absolute error of the angle F_β ≈ 4.065E-6°
• Error of the chord A₁G₁

A₁G₁ is the constructed chord S_2K = 1.99868668102401 [unit of length]

Chord S_2SHOULD = ${\displaystyle \sin \left({\frac {\beta }{2}}\right)\cdot 2}$ = 1.99868667845274 [unit of length]

Absolute error of the chord F_S2 = S_2K - S_2SHOULD = 2.57...E-9 [unit of length]

• If the radius r = 1,000 km would be the error of the chord approx. 2.6 mm.

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