How do calculators evaluate inverse trig functions? I know for simple inputs, you can just memorize the answer, but what if I wanted to find $\arcsin{0.554}$. My calculator instantly tells me that the answer is $0.5752 \ \text{radians}$. How can I do that by hand, procedurally, to always arrive at the right answer without having to memorize a bunch of stuff.
Thanks. 
 A: The calculators normally use a version of the CORDIC-algorithms which is a set of algorithms for evaluating different trigonometric functions, or some kind of power series (e.g. Taylor series) of the function. These algorithms normally are not very well suited for doing calculations 'by hand' since they make use of a lot of repetition where computers are good at but we humans are pretty slow. As the hardware became cheaper we now often also have lookup tables which enables us to do a interpolation which is accurate enough for the expected applications.
A: To answer the question in the body, which is quite different from the title, you won't get a very accurate value for arcsin by hand without a lot of work.  The easiest reasonably accurate way is to use the closest value you know and a Taylor's series.  If you want $\arcsin 0.554$, you probably know $\arcsin 0.5=\frac \pi 6$.  Then $\arcsin 0.554 \approx \frac \pi 6 + 0.054 \left.\frac d{dx}\arcsin x\right|_{0.5}=\frac \pi 6 + 0.054 \frac 1{\sqrt {1-0.5^2}}\approx 0.58595$, which is within $0.0012$ of the correct $0.58716.$
A: It isn't actually instantaneous, but it's quite fast. There are a few ways to estimate functions. One way is to use a Taylor series expansion.
As for doing this by hand, even if you had the taylor series expansion, it would be difficult to do since you're talking about a lot of calculations. Sure, it could be done, but it would be far easier to just use a calculator.
A: I am afraid you won't find an easy by-hand solution. The required numerical methods will involve either sufficiently many tabulated numbers, or substantial arithmetic computation.
The Taylor series does not require any table.
If you can content yourself with a crude approximation, then you can use 
$$\sin x\approx\frac{4x(\pi-x)}{\pi^2},$$ for $x$ in range $[0,\pi]$. The formula is exact for $0$, $\pi/2$ and $\pi$.
Inverting the latter,
$$\arcsin x\approx\frac{\pi}2(1-\sqrt{1-x}).$$
Unfortunately, this uses a square root.
A more accurate result is given by the first two terms of Taylor
$$\arcsin x\approx x+\frac{x^3}6.$$
Use it for $0<x<\sqrt2/2$, and for $x>\sqrt 2/2$, transform as $$\arcsin x=\frac\pi2-\arcsin\sqrt{1-x^2}.$$
I guess it is hard to avoid a square root, as the derivative becomes infinite at $x=1$.
A: This is a good question! Trig functions and inverse trig functions are tricky. In fact, IEEE has published standards for how they should be computed. (EDIT: Clause 9 of IEEE 754-2008, the floating point standard, recommends but does not require the implementation of trigonometric functions.)
In the following paper, a method is described for computing $\arcsin x$ when $0 < x < 1$: Paper.
In essence, they use the trig identity
$$\arcsin N = \arctan \frac{N}{\sqrt{1-N^2}}.$$
Of course, this just raises the question: how do you compute $\arctan x$?
Old algorithms used something called CORDIC.
Modern computers have sufficient memory and speed that they can construct lookup tables and interpolate. However, CORDIC is still finding use in things like FPGAs. I don't know offhand what the standard algorithm is, currently, but I'm willing to bet your calculator either uses CORDIC or interpolated lookups.
