Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/5/84; site anasazi.UUCP Path: utzoo!linus!decvax!ucbvax!ucdavis!lll-crg!seismo!hao!noao!terak!mot!anasazi!steve From: steve@anasazi.UUCP (Steve Villee) Newsgroups: net.micro Subject: Re: Series for Pi and Other Trig Oddities Message-ID: <456@anasazi.UUCP> Date: Mon, 30-Dec-85 14:33:00 EST Article-I.D.: anasazi.456 Posted: Mon Dec 30 14:33:00 1985 Date-Received: Wed, 1-Jan-86 04:23:02 EST References: <1043@brl-tgr.ARPA> Organization: Anasazi, Phoenix Az. Lines: 56 > pi/4 = 4*arctan(1/5) - arctan(1/239) > > I am FASCINATED by the fact that this equality holds. What > is special about these two angles to give such an EXACT result. > > cos(20degrees)*cos(40degrees)*cos(80degrees) = 1/8 > > Does anyone have mathematical insight as to why these things > are true? From the definitions, cos(x) = (e**(x*i) + e**(-x*i)) / 2 sin(x) = (e**(x*i) - e**(-x*i)) / (2*i) tan(x) = sin(x)/cos(x) we can derive cos(-x) = cos(x), sin(-x) = - sin(x), tan(-x) = - tan(x) cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y) sin(x+y) = cos(x)*sin(y) - cos(y)*sin(x) tan(x+y) = (tan(x) + tan(y)) / (1 - tan(x)*tan(y)) Now let a = arctan(1/5), b = arctan(1/239). tan(a) = 1/5 tan(2*a) = 5/12 tan(4*a) = 120/119 tan(b) = 1/239 tan(4*a - b) = 1 4*a - b = pi/4, since a and b are between 0 and pi/4. For the other one, let r = e**(2*pi*i/9). Then r is a primitive ninth root of 1 (that is, r**9 = 1, but this is not true for any smaller positive exponent). cos(20degrees) = - cos(160degrees) = - (r**4 + r**5) / 2 cos(40degrees) = (r + r**8) / 2 cos(80degrees) = (r**2 + r**7) / 2 Expanded out, the product of the three is r**7 + r**8 + r**5 + r**6 + r**3 + r**4 + r + r**2 - -------------------------------------------------- 8 The numerator is just the sum of all the ninth roots of 1 except 1 itself. The sum of all nine roots is 0 (the r**8 term in "r**9 - 1" is 0), so the numerator must be -1. So the product is 1/8. There is probably an easier way to show this, but I have a particular fascination for roots of 1. --- Steve Villee (ihnp4!terak!anasazi!steve) International Anasazi, Inc. 7500 North Dreamy Draw Drive, Suite 120 Phoenix, Arizona 85020 (602) 870-3330