Answer
$2.782369488$
Work Step by Step
With $y=0.35154709$ and $x=-0.9361702$, then
\begin{align*}
\tan{\theta}&=\dfrac{0.35154709}{-0.9361702}\\\\
\theta&=\tan^{-1}\left(\dfrac{0.35154709}{-0.9361702}\right)\\\
\theta&=-0.3592231654
\end{align*}
This angle is in Quadrant IV.
Note that the given point is in Quadrant II.
Since $\tan{\theta}=\tan{(\theta+\pi)}$, then the value of $\theta$ in Quadrant II must be:
\begin{align*}
\theta&=-0.3592231654+\pi\\
&=2.782369488
\end{align*}
Solve for the arc length using the formula $S=r\theta$ with $r=1$ and $\theta=2.782369488$ to obtain:
\begin{align*}
S&=r\theta\\
&=1(2.782369488)\\
&=2.782369488
\end{align*}
Therefore, the length of the shortest arc from $(1, 0)$ to the given point is $2.782369488$.
