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Beliebt Trigonometrie >

sec^2(x)= 1/2 csc^2(x)+tan^2(x)

Lösung

sec^{2} (x) = \frac{1}{2} csc^{2} (x) + tan^{2} (x)

Lösung

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Grad

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

sec^{2} (x) = \frac{1}{2} csc^{2} (x) + tan^{2} (x)

Subtrahiere

\frac{1}{2} csc^{2} (x) + tan^{2} (x)

von beiden Seiten

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x) = 0

Vereinfache

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x) : \frac{2 sec ^{2} ( x ) - csc ^{2} ( x ) - 2 tan ^{2} ( x )}{2}

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x)

\frac{1}{2} csc^{2} (x) = \frac{csc ^{2} ( x )}{2}

\frac{1}{2} csc^{2} (x)

Multipliziere Brüche:

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

= \frac{1 \cdot csc ^{2} ( x )}{2}

Multipliziere:

1 \cdot csc^{2} (x) = csc^{2} (x)

= \frac{csc ^{2} ( x )}{2}

= sec^{2} (x) - \frac{csc ^{2} ( x )}{2} - tan^{2} (x)

Wandle das Element in einen Bruch um:

sec^{2} (x) = \frac{sec ^{2} ( x ) 2}{2}, tan^{2} (x) = \frac{tan ^{2} ( x ) 2}{2}

= \frac{sec ^{2} ( x ) \cdot 2}{2} - \frac{csc ^{2} ( x )}{2} - \frac{tan ^{2} ( x ) \cdot 2}{2}

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

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

= \frac{sec ^{2} ( x ) \cdot 2 - csc ^{2} ( x ) - tan ^{2} ( x ) \cdot 2}{2}

\frac{2 sec ^{2} ( x ) - csc ^{2} ( x ) - 2 tan ^{2} ( x )}{2} = 0

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

2 sec^{2} (x) - csc^{2} (x) - 2 tan^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- csc^{2} (x) + 2 sec^{2} (x) - 2 tan^{2} (x)

Verwende die Pythagoreische Identität:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - csc^{2} (x) + 2 sec^{2} (x) - 2 (sec^{2} (x) - 1)

Vereinfache

- csc^{2} (x) + 2 sec^{2} (x) - 2 (sec^{2} (x) - 1) : - csc^{2} (x) + 2

- csc^{2} (x) + 2 sec^{2} (x) - 2 (sec^{2} (x) - 1)

Multipliziere aus

- 2 (sec^{2} (x) - 1) : - 2 sec^{2} (x) + 2

- 2 (sec^{2} (x) - 1)

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = - 2, b = sec^{2} (x), c = 1

= - 2 sec^{2} (x) - (- 2) \cdot 1

Wende Minus-Plus Regeln an

- (- a) = a

= - 2 sec^{2} (x) + 2 \cdot 1

Multipliziere die Zahlen:

2 \cdot 1 = 2

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

= - csc^{2} (x) + 2 sec^{2} (x) - 2 sec^{2} (x) + 2

Addiere gleiche Elemente:

2 sec^{2} (x) - 2 sec^{2} (x) = 0

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

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

2 - csc^{2} (x) = 0

Löse mit Substitution

2 - csc^{2} (x) = 0

Angenommen:

csc (x) = u

2 - u^{2} = 0

2 - u^{2} = 0 : u = 2, u = - 2

2 - u^{2} = 0

Verschiebe

2

auf die rechte Seite

2 - u^{2} = 0

Subtrahiere

2

von beiden Seiten

2 - u^{2} - 2 = 0 - 2

Vereinfache

- u^{2} = - 2

- u^{2} = - 2

Teile beide Seiten durch

- 1

- u^{2} = - 2

Teile beide Seiten durch

- 1

\frac{- u ^{2}}{- 1} = \frac{- 2}{- 1}

Vereinfache

u^{2} = 2

u^{2} = 2

Für

x^{2} = f (a)

sind die Lösungen

x = f (a), - f (a)

u = 2, u = - 2

Setze in

u = csc (x)

ein

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = 2

Allgemeine Lösung für

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) = - 2

Allgemeine Lösung für

csc (x) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Kombiniere alle Lösungen

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Graph

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