We want to divide the circle into three equal area parts using two chords starting from a point on the circumference of the circle. Determine the approximate value of the angle between the two chords based on the function tables available in school textbooks.

---

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Let the center of the given circle $k$ be $O$, and its radius 1. The chords we are looking for are $AB$ and $AC$, and the angle between them is $x$.

![](https://cdn.mathpix.com/cropped/2024_05_02_955d093f30154d2c82ceg-1.jpg?height=593&width=497&top_left_y=224&top_left_x=814)

The two chords are equal because the extreme circular segments can only be equal in this way; indeed, as the chord increases, the area of the smaller circular segment cut off also increases. Therefore, it is sufficient to determine $x$ such that the sum of the area of the kite $O B A C$ and the area of the circular segment $O B C$ is equal to one-third of the area of $k$.

The perpendicular height of the triangle $O A B$ from $O A$ is $\sin x$, because $\angle B O C = 2x$, and the area of the circular segment is $x$ (in radians), so we need to find the approximate solution to the equation

$$
x + \sin x = \pi / 3 \approx 1.0472
$$

It is easily seen that $x > \pi / 6 (=30^\circ)$, and it only slightly exceeds this limit, because when $x = 30^\circ$, the circular segment is $1/6$ of the area of $k$, and the area of the kite is equal to the area of the inscribed kite $O B D C$ in the circular segment, so the middle part is less than one-third of $k$ by only the segments outside the chords $D B$ and $D C$.

Indeed, when $x = 31^\circ = 0.5411$, $\sin x = 0.5150$, and the left side of (1) is 1.0561, which is 0.0089 more than the right side. Since when $x = 30^\circ = \pi / 6$, the left side is still smaller by $\pi / 6 - 0.5 = 0.0236$, the increase per degree is 0.0325. Assuming the increase in $\sin x$ is uniform over this small interval, we can take $30 + 0.0236 / 0.0325 = 30.73^\circ$ as a new approximate value, i.e., $x = 30^\circ 43.8'$.

Further refinement based on our table is not possible because the increase in the data of our four-digit sine table is uniform in the interval from $30^\circ$ to $31^\circ$ due to rounding.

Döme Éva (Makó, József A. g. III. o. t.)

Note. According to a seven-decimal table, with arc-second precision, $x = 30^\circ 43.55' = 30^\circ 43' 33''$, so the difference is $1/4$ arc-second, which is $1/24$ of the step size of our table. This can be attributed to the fortunate combination of rounding errors.