To make life simpler, we can first restrict the range to the case where the value lies between 0 and 1 by using the identities and . Note that these formulae, and the ones to follow all give the arc tangent in radians so, if you want degrees, you will have to convert the answer. If we only need limited accuracy, a rational approximation is the way to go. The following formula is accurate to two decimal places: Converting to degrees, we have the following: which gives the answer to within a degree. If we want five-place accuracy, there is the following polynomial approximation: This should be good enough for all but the most demanding applications but, if we want arbitrary precision, we could use the following series expansion which has nice convergence properties throughout the range of : Also, for the purpose of numerical computation, this series folds up quite nicely: Because the terms in this series are all positive, it has the "Price is Right" property --- it will estimate the value as closely as you want without ever going over. Let's illustrate this by computing the arc tangent of -3. Using our identities, we see that so we'll first compute the arc tangent of a third, then go back and turn that into the arctangent of minus three. (See the other thread for a debate over whether I should have said "negative three" instead. ) By our rough-and-dirty approximation: or, in degrees, Using the polynomial approximation, Finally, the series becomes As we see from the powers of a tenth, we will gain at least a decimal place of accuracy for each extra term we include. Computing the successive terms numerically, we get the following approximations. 0.300,000,000 0.320,000,000 0.321,600,000 0.321,737,142 0.321,749,333 0.321,750,441 0.321,750,543 . . . . . . . So we're good to seven decimal places so far and have our best value so far, . In particular, comparing with this, we see that the polynomial approximation is good to five decimal places, as advertised. Subtracting the pi, we get Converting into degrees, or, in the good old Babylonian notation, Alright, lesson over, now go compute some angles