Zusammenhang Winkelfunktion Exponentialfunktion

Question

Aufgabe:

Zeigen Sie, dass für z, w ∈ C und t ∈ R gilt:

Text erkannt:

\( \left|e^{i t}-1\right|=2\left|\sin \left(\frac{t}{2}\right)\right| \)

Laut Satz \( 10.5 \mathrm{a})(i): e^{i z}=\cos z+i \sin z \)
\( |\cos t+i \sin t-1|=2\left|\sin \left(\frac{t}{2}\right)\right| \)

Laut Def. 5.5 (ii): \( |z|:=\sqrt{a^{2}+b^{2}} \)
\( \left|\cos t+\sqrt{a^{2} 16^{2}} \cdot \sin t-1\right|=2\left|\sin \left(\frac{t}{2}\right)\right| \)

Lavt \( c_{1}(i v) \cos t=1-2 \sin ^{2}\left(\frac{z}{2}\right) \)
\( \begin{array}{l} \left|1-2 \sin ^{2}\left(\frac{t}{2}\right)+i \cdot \sin t-1\right|=2\left|\sin \left(\frac{t}{2}\right)\right| \\ \left|-2 \sin ^{2}\left(\frac{t}{2}\right)+i \cdot \sin t\right|=2\left|\sin \left(\frac{t}{2}\right)\right| \\ \left|-4 \sin ^{4}\left(\frac{t}{2}\right)-\sin ^{2} t\right|=4\left|\sin ^{2}\left(\frac{t}{2}\right)\right| \\ \left|-2 \sin ^{2}\left(\frac{t}{2}\right) \cdot 2 \sin ^{2}\left(\frac{t}{2}\right)-\sin ^{2} t\right|=4\left|\sin ^{2} \frac{t}{2}\right| \end{array} \)
\( (C)^{2} \)

Problem/Ansatz:

Kann mir jemand weiterhelfen?

Answer 1

\( |\cos t+i \sin t-1|=2\left|\sin \left(\frac{t}{2}\right)\right| \)

<=> \( |\cos t - 1  +i \sin t|=2\left|\sin \left(\frac{t}{2}\right)\right| \)

<=> \(  \sqrt{ (\cos t - 1) ^2 + \sin^2 t }=2\left|\sin \left(\frac{t}{2}\right)\right| \)

alles nicht negativ, da kann ma quadrieren

<=> \(   (\cos t - 1) ^2 + \sin^2 t =4\sin^2 \left(\frac{t}{2}\right) \)

<=> \(  \cos^2 t -2\cos t + 1 + \sin^2 t =4\sin^2 \left(\frac{t}{2}\right) \)

<=> \(  1 -2\cos t + 1 =4\sin^2 \left(\frac{t}{2}\right) \)

<=> \(  2 -2\cos t  =4\sin^2 \left(\frac{t}{2}\right) \)  |:2

<=> \(  1 -\cos t =2\sin^2 \left(\frac{t}{2}\right) \)

<=> \(  1 -\cos (\frac{t}{2}+\frac{t}{2})  =2\sin^2 \left(\frac{t}{2}\right) \)

Additionstheorem anwenden

<=> \(  1 -  (\cos^2 (\frac{t}{2}) -\sin^2(\frac{t}{2}) ) =2\sin^2 \left(\frac{t}{2}\right) \)

<=> \(  1 - \cos^2 (\frac{t}{2})  + \sin^2(\frac{t}{2}) ) =2\sin^2 \left(\frac{t}{2}\right) \)

<=> \(    \sin^2 (\frac{t}{2})  + \sin^2(\frac{t}{2}) ) =2\sin^2 \left(\frac{t}{2}\right) \)

                          BINGO!

Answer 2

Vielleicht noch eine kürzere Lösung:

$$|\exp(it)-1|=|\exp(0.5it)||\exp(0.5it)-\exp(-0.5it)|=1 \cdot |2\sin(0.5t)|$$

Die Umformung der beiden Faktoren folgt jeweils aus Satz 105, wie von FS zitiert.