De Moivre Theorem (1 Viewer)

nrumble42 · Nov 3, 2016

Any help on this would be great guys. It says to simplify, leaving in exact values.

He-Mann · Nov 3, 2016

Paradoxica · Nov 3, 2016

nrumble42 said:
View attachment 33566

Any help on this would be great guys. It says to simplify, leaving in exact values.

$$Observe that $|z| = 1$, where $z = \cos{\frac{\pi}{7}} + i\sin{{\frac{\pi}{7}}} \\\\ $Replace $1$ with $\sqrt{z} \sqrt{\bar{z}}$, giving: $\\\\ \left(\frac{\sqrt{z}\sqrt{\bar{z}} - \bar{z}}{\sqrt{z}\sqrt{\bar{z}} - z} \right)^7 = \left(\frac{\sqrt{\bar{z}}}{\sqrt{z}}\right)^7 \left(\frac{\sqrt{z} - \sqrt{\bar{z}}}{\sqrt{\bar{z}} - \sqrt{z}}\right)^7 = \left( \frac{\bar{z}}{\sqrt{z \bar{z}}} \right)^7 (-1)^7 \\\\ = \left( \cos{\frac{\pi}{7}} - i\sin{{\frac{\pi}{7}}} \right)^7 (-1) = (\cos{\pi} - i \sin{\pi}) (-1) = 1$

De Moivre Theorem (1 Viewer)

nrumble42

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He-Mann

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Paradoxica

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