File:01-Winkel 45,371° 134,629°.gif

Description01-Winkel 45,371° 134,629°.gif	Deutsch: Näherungskonstruktion des Winkels 45,371° mit Supplementwinkel 134,629°, Animation English: Proximity construction of the angle 45,371° with supplementary angle 134,629°, animation
Date	1 November 2016
Source	Own work
Author	Petrus3743
Other versions

• 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.

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

• Absoluter Winkelfehler F_α ≈ 5,526E-10°
• Fehler der Sehne A₁E₁

A₁E₁ ist die konstruierte Sehne S_1K = 0,771345121459908 [LE]

Sehne S_1SOLL = ${\displaystyle \sin \left({\frac {\alpha }{2}}\right)\cdot 2}$ = 0,771345121451010 [LE]

Absoluter Fehler der Sehne F_S1 = S_1K - S_1SOLL = 8,897...E-12 [LE]

• Bei einem Radius r = 1 Mio. km wäre der Fehler an der Sehne ca. 8,9 mm.

• Absoluter Winkelfehler F_β ≈ -5,53-10°
• Fehler der Sehne A₁G₁

A₁G₁ ist die konstruierte Sehne S_2K = 1,84527144442220 [LE]

Sehne S_2SOLL = ${\displaystyle \sin \left({\frac {\beta }{2}}\right)\cdot 2}$ = 1,84527144442592 [LE]

Absoluter Fehler der Sehne F_S2 = S_2K - S_2SOLL = -3,719...E-12 [LE]

• Bei einem Radius r = 1 Mio. km wäre der Fehler an der Sehne ca. -3,7 mm.

• 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.

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

• Absolute error of the angle F_α ≈ 5.526E-10°
• Error of the chord A₁E₁

A₁E₁ is the constructed chord S_1K = 0.771345121459908 [unit of length]

Chord S_1SHOULD = ${\displaystyle \sin \left({\frac {\alpha }{2}}\right)\cdot 2}$ = 0.771345121451010 [unit of length]

Absolute error of the chord F_S1 = S_1K - S_1SHOULD = 8.897...E-12 [unit of length]

• If the radius r = 1 million km would be the error of the chord approx. 8.9 mm.

• Absolute error of the angle F_β ≈ -5.53-10°
• Error of the chord A₁G₁

A₁G₁ is the constructed chord S_2K = 1.84527144442220 [unit of length]

Chord S_2SHOULD = ${\displaystyle \sin \left({\frac {\beta }{2}}\right)\cdot 2}$ = 1.84527144442592 [unit of length]

Absolute error of the chord F_S2 = S_2K - S_2SHOULD = -3.719...E-12 [unit of length]

• If the radius r = 1 million km would be the error of the chord approx. -3.7 mm.

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