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Beliebt >

(sin(x)+cos(x))/(cos(x)+1)=tan(x)

Lösung

\frac{sin ( x ) + cos ( x )}{cos ( x ) + 1} = tan (x)

x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

Schritte zur Lösung

\frac{sin ( x ) + cos ( x )}{cos ( x ) + 1} = tan (x)

Subtrahiere

tan (x)

von beiden Seiten

\frac{sin ( x ) + cos ( x )}{cos ( x ) + 1} - tan (x) = 0

Vereinfache

\frac{sin ( x ) + cos ( x )}{cos ( x ) + 1} - tan (x) : \frac{sin ( x ) + cos ( x ) - tan ( x ) ( cos ( x ) + 1 )}{cos ( x ) + 1}

\frac{sin ( x ) + cos ( x ) - tan ( x ) ( cos ( x ) + 1 )}{cos ( x ) + 1} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (x) + cos (x) - tan (x) (cos (x) + 1) = 0

Drücke mit sin, cos aus

sin (x) + cos (x) - \frac{sin ( x )}{cos ( x )} (cos (x) + 1) = 0

Vereinfache

sin (x) + cos (x) - \frac{sin ( x )}{cos ( x )} (cos (x) + 1) : \frac{cos ^{2} ( x ) - sin ( x )}{cos ( x )}

\frac{cos ^{2} ( x ) - sin ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos^{2} (x) - sin (x) = 0

Füge

sin (x)

zu beiden Seiten hinzu

cos^{2} (x) = sin (x)

Quadriere beide Seiten

(cos^{2} (x))^{2} = sin^{2} (x)

Subtrahiere

sin^{2} (x)

von beiden Seiten

cos^{4} (x) - sin^{2} (x) = 0

Faktorisiere

cos^{4} (x) - sin^{2} (x) : (cos^{2} (x) + sin (x)) (cos^{2} (x) - sin (x))

(cos^{2} (x) + sin (x)) (cos^{2} (x) - sin (x)) = 0

Löse jeden Teil einzeln

cos^{2} (x) + sin (x) = 0 or cos^{2} (x) - sin (x) = 0

cos^{2} (x) + sin (x) = 0 : x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

cos^{2} (x) - sin (x) = 0 : x = arcsin (\frac{5 - 1}{2}) + 2 πn, x = π - arcsin (\frac{5 - 1}{2}) + 2 πn

Kombiniere alle Lösungen

x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = arcsin (\frac{5 - 1}{2}) + 2 πn, x = π - arcsin (\frac{5 - 1}{2}) + 2 πn

Verifiziere Lösungen, indem du sie in die Original-Gleichung einsetzt

x = arcsin (\frac{5 - 1}{2}) + 2 πn, x = π - arcsin (\frac{5 - 1}{2}) + 2 πn

Zeige Lösungen in Dezimalform

x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

cos (x) - cos (2 x) = 1

tan (θ) = \frac{1}{2}

- cos (x) - sin (x) = 1

sin (2 x) = sin (0.5 x)

2 sin (2 x + 15) = \frac{1}{2}