Summary edit

Description01-Winkel 4,153° 175,847°.svg	Deutsch: Näherungskonstruktion des Winkels 4,153° mit Supplementwinkel 175,847° English: Proximity construction of the angle 4.153° with supplementary angle 175.847°
Date	23 October 2016
Source	Own work
Author	Petrus3743
Other versions
SVG development InfoField	The SVG code is valid.   This trigonometry was created with GeoGebra by Petrus3743.   This SVG trigonometry uses the path text method.

Konstruktionsprinzip edit

• Einheitskreis Radius r = 1, in GeoGebra,
• Näherungskonstruktion mittels des dritten Strahlensatzes, siehe hierzu auch Konstruktion einer Dezimalzahl; 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 Konstruktion einer Dezimalzahl; 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.

Licensing edit

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