It’s time for you to check Anita’s work. What was the measure of angle $\varphi$?
Ok, so as we saw in the prior Classic Fiddler problem, Anita will be crossing the $1$-inch segment while she is on her $4$-inch line segment, that is when she is traveling from $$X_3 = (1+2\cos \varphi+3\cos 2\varphi, 2\sin \varphi + 3\sin 2\varphi)$$ to $$X_4 = (1 + 2 \cos \varphi + 3 \cos 2\varphi + 4\cos 3\varphi, 2 \sin \varphi + 3 \sin 2\varphi + 4 \sin 3 \varphi).$$ We have already somewhat loosely located $\varphi \in ( \frac{3\pi}{4}, \frac{4\pi}{5} ).$
We see that the line from $X_3$ to $X_4$ is given by $$y - 2 \sin \varphi - 3 \sin 2\varphi = \tan 3\varphi \left( x - 1 - 2 \cos \varphi - 3 \cos 2\varphi \right).$$ Additionally we note that since for large angles (e.g., $\phi = \pi$) the crossing point is near $(0,0)$ and since the crossing point for the smaller $\phi = \frac{4\pi}{5}$ being closer towards the $(1,0)$ endpoint of the initial segment, it stands to reason that for the optimal angle $\varphi,$ that the crossing point would be precisely at the endpoint $(1,0).$ In this cas we would have $$ - 2 \sin \varphi - 3 \sin 2 \varphi = \tan 3\varphi \left( -2 \cos \varphi - 3 \cos 2\varphi \right).$$ Since $$\tan 3 \varphi = \frac{ 3 \tan \varphi - \tan^3 \varphi }{ 1 - 3 \tan^2 \varphi } = \tan \varphi \frac{ 3 - \tan^2 \varphi }{ 1 - 3 \tan^2 \varphi } = \tan \varphi \frac{ 4 \cos^2 \varphi - 1}{ 4 \tan^2 \varphi - 3},$$ $$-2 \cos \varphi - 3 \cos 2\varphi = 3 - 2 \cos \varphi - 6 \cos^2 \varphi,$$ and $$2 \sin\varphi + 3 \sin 2\varphi = 2 \tan \varphi \cos \varphi ( 1 + 3 \cos \varphi ),$$ the above equality is equivalent (which much algebraic expansion and reduction) to \begin{align*} 0 &= \tan \varphi \frac{4 \cos^2 \varphi - 1}{4 \cos^2 \varphi - 3} ( 3 - 2\cos \varphi - 6 \cos^2 \varphi) + 2 \tan \varphi \cos \varphi (1 + 3 \cos \varphi) \\ &= \frac{\tan \varphi}{4 \cos^3 \varphi - 3} \left( (4 \cos^2 \varphi - 1) ( 3 - 2 \cos\varphi - 6 \cos^2 \varphi ) + 2 \cos \varphi ( 4 \cos^2 \varphi - 3 ) (1 + 3 \cos \varphi) \right) \\ &= - \frac{\tan \varphi (4 \cos \varphi + 3)}{4 \cos^2 \varphi - 3},\end{align*} so we have that $\varphi = \cos^{-1} \left(-\frac{3}{4}\right) \approx 2.41885840578\dots.$