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Beliebt Trigonometrie >

3(cos^2(a))/(sin^2(a))=(cot(a))^2

Lösung

3 \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

Lösung

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Grad

a = 9 0^{\circ} + 36 0^{\circ} n, a = 27 0^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

3 \cdot \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

Subtrahiere

(cot (a))^{2}

von beiden Seiten

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a) = 0

Vereinfache

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a) : \frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a)

Wandle das Element in einen Bruch um:

cot^{2} (a) = \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

= \frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

\frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )} = 0

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

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a) = 0

Faktorisiere

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a) : (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a)

Schreibe

cot^{2} (a) sin^{2} (a)

um:

(cot (a) sin (a))^{2}

cot^{2} (a) sin^{2} (a)

Wende Exponentenregel an:

a^{m} b^{m} = (ab)^{m}

cot^{2} (a) sin^{2} (a) = (cot (a) sin (a))^{2}

= (cot (a) sin (a))^{2}

= 3 cos^{2} (a) - (cot (a) sin (a))^{2}

Schreibe

3 cos^{2} (a) - (cot (a) sin (a))^{2}

um:

(3 cos (a))^{2} - (cot (a) sin (a))^{2}

3 cos^{2} (a) - (cot (a) sin (a))^{2}

Wende Radikal Regel an:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} cos^{2} (a) - (cot (a) sin (a))^{2}

Wende Exponentenregel an:

a^{m} b^{m} = (ab)^{m}

(3)^{2} cos^{2} (a) = (3 cos (a))^{2}

= (3 cos (a))^{2} - (cot (a) sin (a))^{2}

= (3 cos (a))^{2} - (cot (a) sin (a))^{2}

Wende Formel zur Differenz von zwei Quadraten an:

x^{2} - y^{2} = (x + y) (x - y)

(3 cos (a))^{2} - (cot (a) sin (a))^{2} = (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

= (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

(3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a)) = 0

Löse jeden Teil einzeln

3 cos (a) + cot (a) sin (a) = 0 or 3 cos (a) - cot (a) sin (a) = 0

3 cos (a) + cot (a) sin (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) + cot (a) sin (a) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (a) 3 + cot (a) sin (a)

Verwende die grundlegende trigonometrische Identität:

cot (x) = \frac{cos ( x )}{sin ( x )}

= cos (a) 3 + \frac{cos ( a )}{sin ( a )} sin (a)

\frac{cos ( a )}{sin ( a )} sin (a) = cos (a)

\frac{cos ( a )}{sin ( a )} sin (a)

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{cos ( a ) sin ( a )}{sin ( a )}

Streiche die gemeinsamen Faktoren:

sin (a)

= cos (a)

= 3 cos (a) + cos (a)

cos (a) + cos (a) 3 = 0

Löse mit Substitution

cos (a) + cos (a) 3 = 0

Angenommen:

cos (a) = u

u + u 3 = 0

u + u 3 = 0 : u = 0

u + u 3 = 0

Faktorisiere

u + u 3 : (1 + 3) u

u + u 3

Klammere gleiche Terme aus

u

= u (1 + 3)

(1 + 3) u = 0

Teile beide Seiten durch

1 + 3

(1 + 3) u = 0

Teile beide Seiten durch

1 + 3

\frac{( 1 + 3 ) u}{1 + 3} = \frac{0}{1 + 3}

Vereinfache

u = 0

u = 0

Setze in

u = cos (a)

ein

cos (a) = 0

cos (a) = 0

cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

cos (a) = 0

Allgemeine Lösung für

cos (a) = 0

cos (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Kombiniere alle Lösungen

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) - cot (a) sin (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) - cot (a) sin (a) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (a) 3 - cot (a) sin (a)

Verwende die grundlegende trigonometrische Identität:

cot (x) = \frac{cos ( x )}{sin ( x )}

= cos (a) 3 - \frac{cos ( a )}{sin ( a )} sin (a)

\frac{cos ( a )}{sin ( a )} sin (a) = cos (a)

\frac{cos ( a )}{sin ( a )} sin (a)

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{cos ( a ) sin ( a )}{sin ( a )}

Streiche die gemeinsamen Faktoren:

sin (a)

= cos (a)

= 3 cos (a) - cos (a)

- cos (a) + cos (a) 3 = 0

Löse mit Substitution

- cos (a) + cos (a) 3 = 0

Angenommen:

cos (a) = u

- u + u 3 = 0

- u + u 3 = 0 : u = 0

- u + u 3 = 0

Faktorisiere

- u + u 3 : (- 1 + 3) u

- u + u 3

Klammere gleiche Terme aus

u

= u (- 1 + 3)

(- 1 + 3) u = 0

Teile beide Seiten durch

- 1 + 3

(- 1 + 3) u = 0

Teile beide Seiten durch

- 1 + 3

\frac{( - 1 + 3 ) u}{- 1 + 3} = \frac{0}{- 1 + 3}

Vereinfache

u = 0

u = 0

Setze in

u = cos (a)

ein

cos (a) = 0

cos (a) = 0

cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

cos (a) = 0

Allgemeine Lösung für

cos (a) = 0

cos (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Kombiniere alle Lösungen

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Kombiniere alle Lösungen

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Graph

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