We read that the length of the segment

$$
\frac{1}{3}+\frac{1}{5}+\frac{1}{51}+\frac{1}{95}
$$

is a good approximate value for the side of a regular 11-sided polygon inscribed in a circle with a radius of one unit. The central angle corresponding to this chord deviates from $360^{\circ} / 11$ by only about $0.5^{\prime \prime}$. Verify this statement by calculating $\sin \pi / 11$ with sufficient accuracy. We can use the expression found in the formula table of the school function table:

$$
\sin x=\frac{x}{1!}-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}-\ldots
$$

where $x$ is the radian measure of the angle, and $n!$ denotes the product of natural numbers from 1 to $n$, and also the following: $\pi=3.141592653589 \ldots$, and $\pi^{2}=9.869604401089 \ldots$. Suggest another method for verification (no need to execute it).
I. The side of a regular 11-gon inscribed in the unit circle is $2 \sin \pi / 11$. The problem claims that the approximate value of expression (1) is close to this, and even closer to the value we get for the chord length when we take the central angle $2 \pi / 11$ increased or decreased by 0.5 arcseconds, i.e., when we replace $\pi / 11$ in the side formula with a value that is $0.25^{\prime \prime}$ larger or smaller. 0.25 arcseconds is approximately $180 \cdot 60 \cdot 60 \cdot 4 \approx 2.6 \cdot 10^{6}$ times smaller than $180^{\circ}$, so its radian measure is approximately $1.2 \cdot 10^{-6}$. According to (2), it is expected that the sine of the modified angle will differ from $\sin \pi / 11$ by a value of this magnitude, so it will be sufficient to determine the sine of $\pi / 11$ and the modified value to 7 decimal places. Whether we need to increase or decrease $\pi / 11$ by $0.25^{\prime \prime}$ will be seen from whether $\sin \pi / 11$ turned out to be smaller or larger in (1). We will calculate the individual terms of (1) and (2) to 8 decimal places. We will always round the positive terms down and the absolute value of the negative terms up, thus obtaining a lower approximation for the expressions to be calculated.

The calculation of the chord $h$ from (1)

$$
\begin{aligned}
& 1 / 3=0.33333333 \\
& 1 / 5=0.2 \\
& 1 / 51=0.01960784 \\
& 1 / 95=\underline{0.01052631} \overline{0.56346748} \text {, so }
\end{aligned}
$$

$$
0.281733741+5 \cdot 42 \cdot 10^{-7}=1+210 \cdot 10^{-7}
\end{gathered}
$$

The second term above is $133 \cdot 10^{-7}$ times the term, rounded up to 8 decimal places as shown next to the term, the third term is $210 \cdot 70^{-7}$ times the term, rounded down to 0, and similarly, the fourth term does not change. The rounding error in each term is again at most $1 \cdot 10^{-8}$, so the seventh decimal place is 7 or 8. This is half the length of the chord corresponding to the central angle increased by $0.5^{\prime \prime}$.

Comparing this with (3), the statement is acceptable, the difference can be somewhat larger than $0.5^{\prime \prime}$, but it is still close to it.

II. We can calculate $\sin (\varepsilon+\pi / 11)$ in a slightly different way, with not much more work, using the addition theorem if we calculate $\sin \varepsilon$ based on (2), as well as the values of $\cos \pi / 11$ and $\cos \varepsilon$ based on the expression

$$
\cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\ldots
$$

Rounded to 8 decimal places, $\sin \varepsilon=0.00000121 ; \cos \varepsilon=1.00000000 ;$ to 5 decimal places

$$
0.95949<\cos \pi / 11<0.95950
$$

so the second term of $\sin (\varepsilon+\pi / 11)$ is $\sin \pi / 11$, and the first term is $\sin \varepsilon \cos \pi / 11=116 \cdot 10^{-8}$, which is equal to the above increase, up to a difference of $1 \cdot 10^{-8}$.

The difference between the central angle $2 \varphi$ corresponding to the chord $h$ and $2 \pi / 11$ can also be determined from the difference of some sufficiently large multiple of them, as in this case the error also increases by the same factor. In principle, it is simple to consider the 11-fold of them, since in this case we compare $\sin 22 \varphi$ to 0; however, it would be enough to consider $11 / 2$-fold of them, i.e., the difference $\sin 11 \varphi-0$. For example, based on the identities

$$
\begin{gathered}
\sin 3 \varphi=3 \sin \varphi-4 \sin ^{3} \varphi=\sin \varphi\left(3-4 \sin ^{2} \varphi\right)=\sin \varphi\left(4 \cos ^{2} \varphi-1\right) \\
\cos 3 \varphi=4 \cos ^{3} \varphi-3 \cos \varphi=\cos \varphi\left(4 \cos ^{2} \varphi-3\right) \\
\sin 2 \varphi=2 \sin \varphi \cos \varphi, \quad \cos 2 \varphi=2 \cos ^{2} \varphi-1
\end{gathered}
$$

$$
\begin{gathered}
\sin 11 \varphi=\sin (9 \varphi+2 \varphi)=\ldots=\sin \varphi\left(1024 \cos ^{10} \varphi-\right. \\
\left.-2304 \cos ^{8} \varphi+1792 \cos ^{6} \varphi-560 \cos ^{4} \varphi+60 \cos ^{2} \varphi-1\right)
\end{gathered}
$$

where from expression (1), by bringing to a common denominator and halving

$$
\sin \varphi=\frac{91}{323}, \quad \cos ^{2} \varphi=1-\frac{91^{2}}{323^{2}}=\frac{414 \cdot 232}{323^{2}}
$$

(i.e., $\sin 11 \varphi$ is a rational number).

Csörgei József (Budapest, Fazekas M. gyak. g. III. o. t.) Draschitz Rudolf (Budapest, Landler J. Gépip. és Híradásip. Techn. III. o. t.) Gegesy Ferenc (Budapest, Móricz Zs. g. II. o. t.)

Remark. The central angle corresponding to the chord $h$ - first its half - can be calculated from the power series

$$
\operatorname{arc} \sin x=x+\frac{1}{2} \cdot \frac{x^{3}}{3}+\frac{3}{2 \cdot 4} \cdot \frac{x^{5}}{5}+\frac{3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{x^{7}}{7}+\frac{3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8} \cdot \frac{x^{9}}{9}+\ldots
$$

at the value $x=h / 2=91 / 323$:

$$
\operatorname{arc} \sin 91 / 323=0.28560057
$$

The excess of this over the above value of $\pi / 11$ is $124 \cdot 10^{-8} \approx 0.254^{\prime \prime}$. However, 6 terms were needed here, and estimating the sum of the omitted further terms is more difficult because every term is positive.

Dombi József (Szeged, Ságvári E. gyak. g. IV. o. t.)