English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Lösung

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

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

Grad

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

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

Subtrahiere

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

von beiden Seiten

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Doppelwinkelidentität:

sin (2 x) = 2 sin (x) cos (x)

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

Vereinfache

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

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

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

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

2^{2} = 4

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

Addiere gleiche Elemente:

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

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

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

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

Löse jeden Teil einzeln

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

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

cos^{2} (x) = 0

Wende Regel an

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

Allgemeine Lösung für

cos (x) = 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}

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

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

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Wende Regel an

x^{n} = 0 \Rightarrow x = 0

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

Kombiniere alle Lösungen

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

\frac{sin ( 7 0 ^{\circ} )}{25} = \frac{sin ( c )}{26}

1.25 sin (3 0^{\circ}) = 1.5 sin (x)

3 sin (3 x) + 3 cos (3 x) = 0

tan (θ) = - \frac{4}{5}, \frac{π}{2} < θ < π

cos (2 x) = sin (x), 0 \leq x < 2 π