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2sin(x)sec(x)-2sqrt(3)sin(x)=0

Lösung

2 sin (x) sec (x) - 23 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 0.95531 \dots + 2 πn, x = 2 π - 0.95531 \dots + 2 πn

Grad

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 305.26438 \dots^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

2 sin (x) sec (x) - 23 sin (x) = 0

Faktorisiere

2 sin (x) sec (x) - 23 sin (x) : 2 sin (x) (sec (x) - 3)

2 sin (x) sec (x) - 23 sin (x)

Klammere gleiche Terme aus

2 sin (x)

= 2 sin (x) (sec (x) - 3)

2 sin (x) (sec (x) - 3) = 0

Löse jeden Teil einzeln

sin (x) = 0 or sec (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Allgemeine Lösung für

sin (x) = 0

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Löse

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sec (x) - 3 = 0 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) - 3 = 0

Verschiebe

3

auf die rechte Seite

sec (x) - 3 = 0

Füge

3

zu beiden Seiten hinzu

sec (x) - 3 + 3 = 0 + 3

Vereinfache

sec (x) = 3

sec (x) = 3

Wende die Eigenschaften der Trigonometrie an

sec (x) = 3

Allgemeine Lösung für

sec (x) = 3

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Kombiniere alle Lösungen

x = 2 πn, x = π + 2 πn, x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Zeige Lösungen in Dezimalform

x = 2 πn, x = π + 2 πn, x = 0.95531 \dots + 2 πn, x = 2 π - 0.95531 \dots + 2 πn

cot (a) sec (a) = cos (a)

4 sin^{2} (x) = 4 cos (x) + 1

2 tan^{2} (x) + 3 tan (x) - 2 = 0

sin (x) = - 3 cos (x)

2 cos (t) \cdot sin (t) = cos (t)