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 December 27th, 2012, 04:13 PM #1 Member   Joined: Oct 2012 Posts: 56 Thanks: 0 angles I was looking at methods of measuring pi arbitrarily and found this gem. $\frac{\pi}4=4\arctan \frac15-\arctan(\frac1{239})$ I thought to myself, hey that 239 sure looks familiar. I didn't understand that 4 in 4arctan(1/5) so I did the double angle thing twice on it. $\frac{2*\tan\theta}{1-\tan\theta^2}$ $\frac{2*\frac15}{1-\frac15^2} --> \frac{2*\frac5{12}}{1-\frac5{12}^2} --> \frac{120}{119}$ Oh, yeah, I know where that came from now. Time to do the angle addition formula. $\frac{tan\theta-tan\phi}{1-tan\theta*tan\phi}$ $\frac{\frac{120}{119}+\frac1{239}}{1-\frac{120}{119}\frac1{239}}=1$ So, I guess this means there are infinite solutions just like this one for measuring pi. $\frac{\frac43-\frac17}{1+\frac43*\frac17}=1$ $2arctan\frac12-arctan\frac17=45^{\circ}$ $2arctan\frac{17}{41}-arctan\frac1{1393}=45^{\circ}$ $2arctan\frac{29}{70}-arctan\frac1{8119}=45^{\circ}$ $2arctan\frac{99}{239}-arctan\frac1{47321}=45^{\circ}$ Lots of 2arctans there, but I think there is only 1 4arctan. I've tried about 50 of the sequences just now. Maybe I'll have better luck if I tackle the 30/60
 December 28th, 2012, 04:07 PM #2 Member   Joined: Oct 2012 Posts: 56 Thanks: 0 Re: angles Blah
 December 28th, 2012, 05:04 PM #3 Member   Joined: Oct 2012 Posts: 56 Thanks: 0 Re: angles Blah

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