Answer
the curve:
$$ r=\tan \theta $$
is completely traced with
$$
\theta =\frac{\pi}{6} \quad \text {to} \quad \theta= \frac{\pi}{3}
$$
the length of the given curve is equal to
$$
\begin{aligned} L &=\int_{\frac{\pi}{6}}^{\frac{\pi}{3} } \sqrt{r^{2}+(d r / d \theta)^{2}} d \theta \\
\\
& \approx 1.2789
\end{aligned}
$$
Work Step by Step
the curve:
$$ r=\tan \theta $$
is completely traced with
$$
\theta =\frac{\pi}{6} \quad \text {to} \quad \theta= \frac{\pi}{3}
$$
the length of the given curve is equal to
$$
\begin{aligned} L &=\int_{\frac{\pi}{6}}^{\frac{\pi}{3} } \sqrt{r^{2}+(d r / d \theta)^{2}} d \theta \\
\\
& =\int_{\pi / 6}^{\pi / 3} \sqrt{\tan ^{2} \theta+\sec ^{4} \theta} d \theta \\
& \approx 1.2789
\end{aligned}
$$
