Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$ with $z \in \mathbb{C}$.
Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$
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algebra-precalculus
complex-numbers
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0Is there a mistake in this question? Should it be determine z? You seem to be given $z^n+z^{-n}$... – 2011-12-01
2 Answers
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Since $z + \frac{1}{z} = - 2 \cos(x)$ is equivalent to $z^2 + 2 z \cos(x) + 1 = 0$, it is solved by $z_{1,2} = -\cos(x) \pm i \sqrt{1-\cos^2(x)}$. Since $1-\cos^2(x) = \sin^2(x)$, these also solve the equation $\tilde{z}_{1,2} = -\cos(x) \mp i \sin(x) = -\exp(\pm i x)$.
Now to find $z^n+z^{-n}$ for $z$ being the solution of $z+\frac{1}{z} = -2 \cos(x)$ subsitute the $z = \tilde{z}_{1,2}$.
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0You can also figure it out by looking to the starting equation ;) – 2011-11-04
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The hint from user9176 is important: $ z+\frac1z = -2\cos x $ Multiply both sides by $z$: $ z^2 + 1 = -2z\cos x $ That's a quadratic equation in $z$. Solve it.
Then remember certain identities involving $e^{ix}$.