Task A-4.6. (10 points)

Determine all values that the expression

$$
\left|\frac{z}{|z|}+\frac{|z|}{4 z}\right|
$$

can take, where $z$ is a complex number different from zero.
## First Solution.

By converting to the trigonometric form \( z = r(\cos \varphi + i \sin \varphi) \), we get

\[
\begin{aligned}
\left|\frac{z}{|z|} + \frac{|z|}{4z}\right| & = \left|\cos \varphi + i \sin \varphi + \frac{1}{4(\cos \varphi + i \sin \varphi)}\right| \\
& = \left|\cos \varphi + i \sin \varphi + \frac{\cos \varphi - i \sin \varphi}{4}\right| \\
& = \left|\frac{5}{4} \cos \varphi + \frac{3}{4} i \sin \varphi\right| \\
& = \sqrt{\frac{25}{16} \cos^2 \varphi + \frac{9}{16} \sin^2 \varphi} \\
& = \sqrt{\frac{9}{16} + \cos^2 \varphi}.
\end{aligned}
\]

Since \( 0 \leq \cos^2 \varphi \leq 1 \), it follows that

\[
\frac{3}{4} \leq \sqrt{\frac{9}{16} + \cos^2 \varphi} \leq \frac{5}{4}
\]

Therefore,

\[
\left|\frac{z}{|z|} + \frac{|z|}{4z}\right| \in \left[\frac{3}{4}, \frac{5}{4}\right]
\]

Note. After obtaining the expression \(\sqrt{\frac{25}{16} \cos^2 \varphi + \frac{9}{16} \sin^2 \varphi}\), it is possible to express everything in terms of \(\sin \varphi\) instead of \(\cos \varphi\) and obtain the same result.