The plane $S$ forms a $30^{\circ}$ angle with the horizontal. The plane $S$ is rotated $120^{\circ}$ around a vertical line as the axis. What is the angle between the resulting plane and the plane $S$?
Consider the $x, y$-axis coordinate system, which spans the horizontal plane. The plane $S$, which forms a $30^{\circ}$ angle with the coordinate plane, coincides with the $y$-axis.

Front view

![](https://cdn.mathpix.com/cropped/2024_05_02_0b4699ac1c1f491284f3g-1.jpg?height=553&width=562&top_left_y=325&top_left_x=337)

Figure 1

![](https://cdn.mathpix.com/cropped/2024_05_02_0b4699ac1c1f491284f3g-1.jpg?height=653&width=550&top_left_y=907&top_left_x=1208)

Figure 2

To find the dihedral angle between the planes $S$ and $S^{\prime}$, which is rotated by $120^{\circ}$, it is sufficient to determine the dihedral angle between their normal vectors, $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$. When rotated, the projections of the normal vectors onto the $(x, y)$ plane also rotate by $120^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_0b4699ac1c1f491284f3g-1.jpg?height=364&width=437&top_left_y=1691&top_left_x=844)

Figure 3

The planes $S$ and $S^{\prime}$ form a $30^{\circ}$ angle with the $(x, y)$ plane, so their normal vectors form a $60^{\circ}$ angle with the $(x, y)$ plane. Let's draw their positions: (Figure 3).

Let the length of the normal vectors be unit length, i.e., $O A = O B = 1$. From the rotation, $\angle B^{\prime} O A^{\prime} = 120^{\circ}$. In the isosceles triangle $A^{\prime} O B^{\prime}$, $O A^{\prime} = O B^{\prime} = \frac{1}{2}$, and $A A^{\prime} = B B^{\prime} = \frac{\sqrt{3}}{2}$. The midpoint $F$ of segment $A^{\prime} B^{\prime}$ has $O F = \frac{1}{4}$ and $F B^{\prime} = \frac{\sqrt{3}}{4}$.

Due to the rotation, $A B$ is parallel to the $(x, y)$ plane, so $\overline{A B} = \overline{A^{\prime} B^{\prime}} = 2 \overline{F B^{\prime}} = \frac{\sqrt{3}}{2}$. In the isosceles triangle $A O B$, the desired angle $\phi$ is such that $\sin \frac{\phi}{2} = \frac{\sqrt{3}}{4}$, from which $\frac{\phi}{2} = 25.66^{\circ}$ and $\phi = 51.31^{\circ}$.