[VA] SRC#004- Fun with Sexagesimal Trigs
02-24-2019, 06:17 PM (This post was last modified: 02-24-2019 07:15 PM by Valentin Albillo.)
Post: #15
 Valentin Albillo Senior Member Posts: 479 Joined: Feb 2015
RE: [VA] SRC#004- Fun with Sexagesimal Trigs
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Hi, all:

A short proof of the evaluation of B:$B = \frac{1}{sin\;45º \; sin\;46º} +\frac{1}{sin\;47º \; sin\;48º} \; + \;...\; + \; \frac{1}{sin\;133º \; sin\;134º} = \frac{1}{sin\;1º}$1) First, we multiply both sides by sin 1º: $\frac{sin \;1º}{sin\;45º \; sin\;46º} +\frac{sin \;1º}{sin\;47º \; sin\;48º} \; + \;...\; + \; \frac{sin \;1º}{sin\;133º \; sin\;134º} = \frac{sin \;1º}{sin\;1º} = 1$2) Now we use the identity
$\frac{sin((k + 1)º - kº)}{sin\;kº\;sin(k + 1)º} = cot\;kº - cot(k + 1)º$which transforms the left-hand side into this: $cot\; 45º - cot\; 46º + cot\; 47º - cot\; 48º +···+ cot\; 133º - cot\; 134º$3) Then we reorder the terms in the sum like this: $cot \;45º - (cot \;46º + cot \;134º) + (cot \;47º + cot \;133º) - ··· + (cot \;89º + cot \;91º) - cot \;90º$4) All the terms inside the parentheses cancel out because they feature supplementary angles ( cot Nº + cot (180º-Nº) = 0 ), so the expression reduces to: $cot \;45º - cot \;90º = 1 -0 = 1$
Q.E.D.

Thanks for your interest and have a nice weekend.
V
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 Messages In This Thread [VA] SRC#004- Fun with Sexagesimal Trigs - Valentin Albillo - 02-11-2019, 04:13 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - J-F Garnier - 02-11-2019, 07:55 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - J-F Garnier - 02-11-2019, 09:26 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Gerson W. Barbosa - 02-11-2019, 09:51 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Albert Chan - 02-12-2019, 03:37 AM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - J-F Garnier - 02-12-2019, 08:05 AM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Gerson W. Barbosa - 02-12-2019, 09:50 AM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Albert Chan - 02-12-2019, 12:14 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Gerson W. Barbosa - 02-12-2019, 01:23 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - J-F Garnier - 02-12-2019, 02:11 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Albert Chan - 02-12-2019, 04:39 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Albert Chan - 02-12-2019, 05:58 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Carsen - 02-14-2019, 03:33 AM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Albert Chan - 02-14-2019, 01:23 PM RE: [VA] SRC#004- Fun with Sexagesimal Trigs - Valentin Albillo - 02-24-2019 06:17 PM

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