# How to calc arc sine without a calculator?

How can I find the arc sine of a sine without using a calculator? Thank you.

• Draw a unit circle and get out your ruler and protractor. Jun 16, 2013 at 15:58

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 $\sin(x)$.

• I am reading Hodges' book about Alan Turing, and in it, Hodges mentions that Turing as a schoolboy derived an expression for the arctangent in terms of the half-angle formula for tangent. I didn't put 2 and 2 together and was trying to figure out how he did this, and then I see your solution and now know exactly how. Thanks (+1)! Jun 16, 2013 at 15:54
• @RonGordon: it's nice to know that something I've done has enriched someone's day :-)
– robjohn
Jun 16, 2013 at 16:46
• Thanks! That first one sure converges slowly for values near 1, though – e.g. to do arcsin 0.999 to 20 decimal places takes 40,000 terms! It seems that how to use the second method was left as an exercise to the reader so I'll keep trying to make sense of it. Oct 26, 2022 at 4:36

$\arcsin$ 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.

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 ) .