ArcTan without calculator  Printable Version + HP Forums (https://archived.hpcalc.org/museumforum) + Forum: HP Museum Forums (https://archived.hpcalc.org/museumforum/forum1.html) + Forum: Old HP Forum Archives (https://archived.hpcalc.org/museumforum/forum2.html) + Thread: ArcTan without calculator (/thread229210.html) 
ArcTan without calculator  PatrickR  08142012 I had a situation yesterday that I would have solved by whipping out the DM15, but I had forgotten it, so manual calculation was the only option. I must though humbly admit that I utterly failed. We had a BBQ at a friend, who was constructing a new shed. The roof was not yet finished and we starting talking about roofing materials, but then came into that depending on the roof angle, there is a need for an underlayer. And then we wanted to know the roof angle... We got as far as seeing it as a right angled triangle were the base is 3 and the height is 1, the side thus sqrt(10). Thus we only needed to calculate arctan(1/3) to know the roof angle. (I later put this to my neighbour who was a carpenter, he knew by heart several angles that you have when triangle height is 1 and base is an integer). No calculators at the house. No smartphones. Nobody wanted to go in to boot up a computer. Everyone pondered over the evening about how to solve this (series expansions: but who remembered the formula, integrals: something with 1 over a square root of something). In the end, several calculations leading to nowhere, we had to give up and ring one of the kids who opened a Wolfram Alpha tab (what a computation overkill). I don't know the arc trig or the regular trig expansions by heart, was there a way this could be solved logically with precalculus math?
Edited: 14 Aug 2012, 5:44 a.m.
Re: ArcTan without calculator  Gilles Carpentier  08142012 Hi
1/ For 'small' angle in radian, ATAN(a)~=a so here :
a ~ 180/3*7/22 If you take PI=3 (!) , just 180/9 > ~ 20° not too bad 2/ With pen, paper and ruler ;)
Edited: 14 Aug 2012, 6:39 a.m.
Re: ArcTan without calculator  x34  08142012 Sure. x<<1 sin(x)~x, cos(X)~1, tan(x)~x => atan(x)~x Re: ArcTan without calculator  Paul Dale  08142012 My first guess would be about twenty degrees. 1:sqrt(3):2 is a Pythagorean triad for a ninety/sixty/thirty degree triangle and this is a bit less.
Re: ArcTan without calculator  PatrickR  08142012 1) It's so simple... Why couldn't we think of that ! Edited: 14 Aug 2012, 6:41 a.m.
Re: ArcTan without calculator  John Mosand  08142012 A simple approximate solution is that from ratio 1:3 down, the angle is found by 57 (roughly 1 radian) divided by the dominator. E.g. 57/3=19, 57/4=14.25, 57/5=11.4, etc. The smaller the fraction, the more exact result.
Re: ArcTan without calculator  Thomas Klemm  08142012 To improve the accuracy you could use the following formula:
I assume you know how to calculate sqrt(10): 31622 Thus we get:
We end up with:
Compare that to the exact value: 18.4349
Kind regards Re: ArcTan without calculator  Gilles Carpentier  08142012 Even simpler : ArcTan(1/3) ~> 1/3 ~> 0.3333 ~> PI/10 ~> 180°/10 ~> 18° For point 2, i mean a 'rapporteur' but i don't know the english word
Edited: 14 Aug 2012, 12:16 p.m.
Re: ArcTan without calculator  Dave Shaffer (Arizona)  08142012 Quote:
That's a "protractor" (as PatrickR mentioned)
