How to calc arc sine without a calculator? How can I find the arc sine of a sine without using a calculator? Thank you.
 A: To compute $\arcsin(x)$ we might need to use a bit of calculus.
Note that
$$
\arcsin(x)=\int_0^x\frac{\mathrm{d}t}{\sqrt{1-t^2}}
$$
Using the binomial theorem, we get that
$$
\frac1{\sqrt{1-x^2}}=\sum_{k=0}^\infty\binom{2k}{k}\left(\frac{x}{2}\right)^{2k}
$$
Integrating term by term, we get
$$
\arcsin(x)=\sum_{k=0}^\infty\frac2{2k+1}\binom{2k}{k}\left(\frac{x}{2}\right)^{2k+1}
$$

Iterative Method Requiring Square Roots
We can use the identity
$$
\begin{align}
\sin^2(x/2)
&=\frac{1-\sqrt{1-\sin^2(x)}}{2}\\[6pt]
&=\frac{\sin^2(x)}{2+2\sqrt{1-\sin^2(x)}}
\end{align}
$$
and the limit
$$
\lim_{n\to\infty}2^n\sin(x/2^n)=x
$$
to compute $x$ from $\sin(x)$.
A: $\arcsin $ is defined to be the inverse of $\sin$ but restricted to a certain range. Hence $\arcsin(\sin(x))=x$ if $x$ is within this range (generally either $0$ to $2\pi$ or $-\pi$ to $\pi$) or a value $y$ such that $\sin(y)=\sin(x)$ i.e. $y=x+2\pi n$ or $y=\pi -x +2\pi m$ for some $n\in \mathbb{Z}$ or $m\in \mathbb{Z}$ and $y$ is in this range.
A: Make a table of sine function (with a step that you choose small for bigger precision ) and use inverse interpolation ( it is a formula for error of inverse interpolation too see books of numerical analysis) ; A table of logs of trigonometric functions and log of numbers (e.g. Hedrick The macmillan logaritmic and trigonometric tables ) .
