HP 17B Solver - another ARCTAN(Y/X) approximation
08-29-2021, 11:39 PM (This post was last modified: 08-30-2021 12:22 PM by Albert Chan.)
Post: #14
 Albert Chan Senior Member Posts: 2,145 Joined: Jul 2018
RE: HP 17B Solver - another ARCTAN(Y/X) approximation
(05-31-2021 09:51 PM)Albert Chan Wrote:  $$\displaystyle\arctan(x) = 2\arctan\left( {x \over \sqrt{1+x^2}+1} \right)$$

Another proof, using atanh

Let y = (1+x)/(1-x), we have x = (y-1)/(y+1)

atanh(x) = ln(y)/2 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ // definition
= ln(√y)
= 2 * atanh((√y-1)/(√y+1))

XCAS> assume(-1 < x < 1)
XCAS> simplify(subst((√y-1)/(√y+1), y = (1+x)/(1-x)))

(-(sqrt(-x^2+1)) + 1) / x

Multiply by conjugate, we have:

atanh(x) = 2 * atanh(x/(√(1-x^2)+1))
atanh(x*i) = 2 * atanh(x/(√(1+x^2)+1)*i)

Divide both side by i, we have atan(x) = 2 * atan(x/(√(1+x^2)+1))

---

Just for fun, I try same way, but with ln(y) = 3 * ln(³√y).
Convert back to atan, I got:

atan(x) = 3 * atan(x/(2*cos(atan(x)/3)*√(1+x^2)+1))

Formula is useless, because reduced argument also call atan(x) « Next Oldest | Next Newest »

 Messages In This Thread HP 17B Solver - another ARCTAN(Y/X) approximation - Martin Hepperle - 05-15-2021, 09:14 AM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-15-2021, 11:01 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Martin Hepperle - 05-16-2021, 08:42 AM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-16-2021, 01:03 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-16-2021, 05:04 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-17-2021, 12:43 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-31-2021, 04:42 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-31-2021, 07:30 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-31-2021, 09:51 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Martin Hepperle - 06-01-2021, 03:16 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 06-01-2021, 04:38 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 08-29-2021 11:39 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 05-17-2021, 06:01 PM RE: HP 17B Solver - another ARCTAN(Y/X) approximation - Albert Chan - 08-19-2021, 02:42 AM

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