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I know that you can convert $\arctan$ value of say $1$ to degrees using Power series.

Example converting $arctan$ value of $1/1$ to degrees:

\begin{align*} &\quad \arctan(1) \\ &=(\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-...-\frac{1}{999}+\frac{1}{1001}-...-\frac{1}{\infty}+\frac{1}{\infty +2}) \\ & \cdot \frac{180}{\pi} = \end{align*} approaching $45$ degrees ($45$ is limit).

The more additions/substractions the closer one gets to the limit.

Here comes my question, is there a better/different way to calculate $\arcsin,\arctan,\arccos$ without using the Power series as in getting to the limit (which acts as degrees)? If you know please give me a full example without cutting the corners. Examples demonstrate the logic behind a solution better than a short version of theory. Thanks.

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