ARG!
2*acos(-3/8)+1*acos(-23/32) = 360.0
acos(3/8) = atan( sqrt55 / 3 ) = arg( 3 + i q55) ;using q as sqrt sign (3 + i q55) ^2 = -46 + i 6 q55 the norm of 3+iq55 is 8, and the norm of -46+i6q55 is 64. 46/64 = 23/32. Rich ________________________________________ From: math-fun [math-fun-bounces@mailman.xmission.com] on behalf of rwg [rwg@sdf.org] Sent: Thursday, July 16, 2015 4:56 PM To: math-fun Subject: [EXTERNAL] Re: [math-fun] ArcCosine Miracles List, and explanation On 2015-07-16 08:27, Warren D Smith wrote:
It is interesting that some of the most-amazing among the listed miracles involve only powers of 2 in denominators:
5*acos(-7/8)+2*acos(-61/64) =1080.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 ...
Hey, 3-4-5 ! In[905]:= 3*ArcCos@(-17/32) + 4*ArcCos@(-11/16) == 5*Pi In[906]:= FullSimplify[%] Out[906]= 3 \[Pi] == 6 ArcSin[17/32] + 8 ArcSin[11/16] is dumfounded, too. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun