For $z+\frac{1}{z}=2\cos\alpha$ calculate $z^n+\frac{1}{z^n}$ If $n$ is a positive integer

Question

here's my first attempt

$z^2-2\cos\alpha+1=0$ so we have $z=\cos\alpha\pm i\sin\alpha= e^{\pm i\alpha}$

but I thought there is a simple approach

What other approach would you suggest?

This is the proper approach. If that is the value of $z$, what is $z^n$? — Aaron, Dec 27 '20 at 21:08

score 1 · Accepted Answer · answered Dec 27 '20 at 21:14

1

$$\small z+\frac{1}{z}=2\cos{\alpha}\iff\left(z-\cos{\alpha}\right)^{2}+\sin^{2}{\alpha}=0\iff z=\cos{\alpha}\pm\mathrm{i}\sin{\alpha}\iff z=\mathrm{e}^{\pm\mathrm{i}\alpha}\iff z^{n}=\mathrm{e}^{\pm n\mathrm{i}\alpha} $$

Thus : $$ z^{n}+\frac{1}{z^{n}}=\mathrm{e}^{\pm n\mathrm{i}\alpha}+\mathrm{e}^{\mp n\mathrm{i}\alpha}=2\cos{\left(n\alpha\right)} $$

answered Dec 27 '20 at 21:14

CHAMSI

8,333

CHAMSI,So beautiful solution, thanks – user62498 Dec 27 '20 at 21:24

For $z+\frac{1}{z}=2\cos\alpha$ calculate $z^n+\frac{1}{z^n}$ If $n$ is a positive integer

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