erweitern (121e^{2t}sin(t)+121e^{2t}cos(t))^2

Lösung

erweitern

(121 e^{2 t} sin (t) + 121 e^{2 t} cos (t))^{2}

14641 e^{4 t} (1 + sin (2 t))

Schritte zur Lösung

(121 e^{2 t} sin (t) + 121 e^{2 t} cos (t))^{2}

Wende Formel für perfekte quadratische Gleichungen an:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 121 e^{2 t} sin (t), b = 121 e^{2 t} cos (t)

= (121 e^{2 t} sin (t))^{2} + 2 \cdot 121 e^{2 t} sin (t) \cdot 121 e^{2 t} cos (t) + (121 e^{2 t} cos (t))^{2}

Vereinfache

(121 e^{2 t} sin (t))^{2} + 2 \cdot 121 e^{2 t} sin (t) \cdot 121 e^{2 t} cos (t) + (121 e^{2 t} cos (t))^{2} : 14641 e^{4 t} sin^{2} (t) + 29282 e^{4 t} sin (t) cos (t) + 14641 e^{4 t} cos^{2} (t)

= 14641 e^{4 t} sin^{2} (t) + 29282 e^{4 t} sin (t) cos (t) + 14641 e^{4 t} cos^{2} (t)

Verwende die Pythagoreische Identität:

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

= 14641 e^{4 t} (2 cos (t) sin (t) + 1)

Verwende die Doppelwinkelidentität:

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

= 14641 e^{4 t} (1 + sin (2 t))

Faktorisiere

14641 e^{4 t} sin^{2} (t) + 29282 e^{4 t} sin (t) cos (t) + 14641 e^{4 t} cos^{2} (t) : 14641 e^{4 t} (sin^{2} (t) + 2 sin (t) cos (t) + cos^{2} (t))

= 14641 e^{4 t} (sin (2 t) + 1)

s im pl i f y sin (2 t δ (t - 2))

e x p an d \frac{6 s + 2 s ^{2} + 2}{( s + 1 ) ( s ^{2} + 1 )}

e x p an d \frac{3 ( 3 - 3 )}{2}

e x p an d \frac{2 e ^{x} - x e ^{2}}{( 1 - x ) ^{2}}

\frac{d ^{2}}{d x ^{2}} ((f (x, y)) (- 6 x y - 6 x))