vereinfachen 1/((-0.75e^{-ia)+1)(-0.5e^{-ia}+1)}

Lösung

vereinfachen

\frac{1}{( - 0.75 e ^{- ia} + 1 ) ( - 0.5 e ^{- ia} + 1 )}

\frac{1 + 0.375 cos ^{2} ( a ) - 0.375 sin ^{2} ( a ) - 1.25 cos ( a )}{1 + 0.140625 cos ^{4} ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) - 0.9375 cos ^{3} ( a ) + 2.3125 cos ^{2} ( a ) - 2.5 cos ( a ) + 0.28125 cos ^{2} ( a ) sin ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a )} + i \frac{- 1.25 sin ( a ) + 0.75 cos ( a ) sin ( a )}{1 + 0.140625 cos ^{4} ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) - 0.9375 cos ^{3} ( a ) + 2.3125 cos ^{2} ( a ) - 2.5 cos ( a ) + 0.28125 cos ^{2} ( a ) sin ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a )}

Schritte zur Lösung

\frac{1}{( - 0.75 e ^{- ia} + 1 ) ( - 0.5 e ^{- ia} + 1 )}

(- 0.75 e^{- ia} + 1) (- 0.5 e^{- ia} + 1) = (- 0.75 (cos (- a) + i sin (- a)) + 1) (- 0.5 (cos (- a) + i sin (- a)) + 1)

= \frac{1}{( - 0.75 ( cos ( - a ) + i sin ( - a ) ) + 1 ) ( - 0.5 ( cos ( - a ) + i sin ( - a ) ) + 1 )}

(- 0.75 (cos (- a) + sin (- a) i) + 1) (- 0.5 (cos (- a) + sin (- a) i) + 1) : ((- 0.5 cos (a) + 1) (- 0.75 cos (a) + 1) - 0.375 sin^{2} (a)) + i (1.25 sin (a) - 0.75 sin (a) cos (a))

= \frac{1}{( ( - 0.5 cos ( a ) + 1 ) ( - 0.75 cos ( a ) + 1 ) - 0.375 sin ^{2} ( a ) ) + i ( 1.25 sin ( a ) - 0.75 sin ( a ) cos ( a ) )}

Entferne die Klammern:

(- a) = - a

= \frac{1}{- 0.375 sin ^{2} ( a ) + ( - 0.5 cos ( a ) + 1 ) ( - 0.75 cos ( a ) + 1 ) + i ( 1.25 sin ( a ) - 0.75 sin ( a ) cos ( a ) )}

Rationalisiere

\frac{1}{- 0.375 sin ^{2} ( a ) + ( - 0.5 cos ( a ) + 1 ) ( - 0.75 cos ( a ) + 1 ) + i ( 1.25 sin ( a ) - 0.75 sin ( a ) cos ( a ) )} : \frac{0.375 cos ^{2} ( a ) - 1.25 cos ( a ) + i ( 0.75 sin ( a ) cos ( a ) - 1.25 sin ( a ) ) + 1 - 0.375 sin ^{2} ( a )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1}

= \frac{- 0.375 sin ^{2} ( a ) + 1 - 1.25 cos ( a ) + 0.375 cos ^{2} ( a ) + i ( 0.75 sin ( a ) cos ( a ) - 1.25 sin ( a ) )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1}

Schreibe

\frac{- 0.375 sin ^{2} ( a ) + 1 - 1.25 cos ( a ) + 0.375 cos ^{2} ( a ) + i ( 0.75 sin ( a ) cos ( a ) - 1.25 sin ( a ) )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1}

in der Standard komplexen Form um:

\frac{- 0.375 sin ^{2} ( a ) + 1 - 1.25 cos ( a ) + 0.375 cos ^{2} ( a )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1} + \frac{0.75 sin ( a ) cos ( a ) - 1.25 sin ( a )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1} i

= \frac{- 0.375 sin ^{2} ( a ) + 1 - 1.25 cos ( a ) + 0.375 cos ^{2} ( a )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1} + \frac{0.75 sin ( a ) cos ( a ) - 1.25 sin ( a )}{0.140625 cos ^{4} ( a ) - 0.9375 cos ^{3} ( a ) + 0.28125 sin ^{2} ( a ) cos ^{2} ( a ) + 2.3125 cos ^{2} ( a ) - 0.9375 sin ^{2} ( a ) cos ( a ) - 2.5 cos ( a ) + 0.140625 sin ^{4} ( a ) + 0.8125 sin ^{2} ( a ) + 1} i

e x p an d (- 3, t + 2 cos (t) + 1) (- 3 cos (t) - 2, t)

e x p an d \frac{( L + r _{1} r _{2} C ) ( r _{1} + r _{2} )}{C}

e x p an d (k^{7})^{2}

e x p an d r (sin (θ) + cos (θ)) d θ

e x p an d csc (\frac{π}{2} + α) - sec (π - α)