ArcTan without calculator - Printable Version +- HP Forums (https://archived.hpcalc.org/museumforum) +-- Forum: HP Museum Forums (https://archived.hpcalc.org/museumforum/forum-1.html) +--- Forum: Old HP Forum Archives (https://archived.hpcalc.org/museumforum/forum-2.html) +--- Thread: ArcTan without calculator (/thread-229210.html) ArcTan without calculator - PatrickR - 08-14-2012 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. I wonder how I should have done. 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 pre-calculus math? Edited: 14 Aug 2012, 5:44 a.m. Re: ArcTan without calculator - Gilles Carpentier - 08-14-2012 Hi 1/ For 'small' angle in radian, ATAN(a)~=a Pi ~= 22/7 so here : ``` a ~ 180/3*7/22 ~ 60*7/22 ~ 420/22 ~ 210/11 a ~ 19 ° ``` 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 - 08-14-2012 Sure. x<<1| sin(x)~x, cos(X)~1, tan(x)~x => atan(x)~x for atan(1/3) the error is less than 3.6%. Re: ArcTan without calculator - Paul Dale - 08-14-2012 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. - Pauli Re: ArcTan without calculator - PatrickR - 08-14-2012 1) It's so simple... Why couldn't we think of that ! 2) A ruler that can measure degrees you mean (protractor)? Edited: 14 Aug 2012, 6:41 a.m. Re: ArcTan without calculator - John Mosand - 08-14-2012 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 - 08-14-2012 To improve the accuracy you could use the following formula:  I assume you know how to calculate sqrt(10): ``` 31622 10 3 100 61 3900 626 14400 6322 175600 63242 ``` Thus we get:  We end up with:  Compare that to the exact value: 18.4349 Obviously we don't really need four places. Kind regards Thomas Re: ArcTan without calculator - Gilles Carpentier - 08-14-2012 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) - 08-14-2012 Quote:For point 2, i mean a 'rapporteur' but i don't know the english word That's a "protractor" (as PatrickR mentioned)