Question

If Z is a complex number such that $z+\frac{1}{z}=2\cos 3$, then find the value of $z^{2000}+\frac{1}{z^{2000}}+1?$

Answer 1

Let $z=e^{i\theta }$

Therefore, $z+\frac{1}{z}=e^{i\theta }+e^{-i\theta }$

Applying the second concept mentioned above,

$z+\frac{1}{z}=2\cos \left(\theta \right)=2\cos \left(3\right)$

Therefore, $\theta =3$

And $z=e^{i3}$

so, $z^{2000}=e^{i6000}$

Substitute this value in $z^{2000}+\frac{1}{z^{2000}}+1$

and you will get, $e^{i6000}+e^{-i6000}+1$

$e^{i6000}+e^{-i6000}=2cos\left(6000\right)$ by applying the second concept mentioned above.

Therefore the final answer is $2\cos \left(6000\right)+1$