| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
The relationship
Arctan(1) + Arctan(2) + Arctan(3) = pi
fascinates me. Not to get too metamathematical, but why should such a
surprising and pretty relationship exist?
BTW, this relationship is responsible for the quickly converging series
for pi/4:
[1/2 + 1/3] -
[(1/2)^3 + (1/3)^3]/3 +
[(1/2)^5 + (1/3)^5]/5 -
[(1/2)^7 + (1/3)^7]/7 +
[(1/2)^9 + (1/3)^9]/9 -
... +
{ (-1)^(n+1) * [(1/2)^(2n-1) + (1/3)^(2n-1)]/(2n-1) } +
...
n series value %error
= =========== =======
1 0.833333333 6.1033
2 0.779320988 -0.7738
3 0.786394033 0.1268
4 0.785212640 -0.0236
5 0.785435299 0.0047
6 0.785390397 -0.0010
7 0.785399835 0.0002
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 532.1 | Confucious say, "picture=1K words" | MODEL::YARBROUGH | Mon Oct 13 1986 16:48 | 13 | |
Geometrically it's pretty straightforward. Construct the following figure: From to 0,0 0,1 0,1 2,3 2,3 2,0 2,0 0,0 0,1 1,0 1,0 2,3 Now the three angles meeting at the bottom center of this figure can be seen to be ArcTan(1), ArcTan(2), and ArcTan(3), respectively. A variant of this figure can be used to show also that ArcTan(1) = ArcTan(1/2) + ArcTan(1/3) | |||||