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Unformatted text preview: edia.or g /wiki/List_of_tr ig onometr ic_identities 9/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia The fac tors 1, 2, 4, 5, 8, 10 may s tart to mak e the pattern c lear: they are thos e integers les s than 21/2 that are relatively prime to (or have no prime fac tors in c ommon with) 21. The las t
s everal ex amples are c orollaries of a bas ic fac t about the irreduc ible c y c lotomic poly nomials : the c os ines are the real parts of the z eroes of thos e poly nomials ; the s um of the z eroes is
the Möbius func tion evaluated at (in the very las t c as e above) 21; only half of the z eroes are pres ent above. The two identities prec eding this las t one aris e in the s ame fas hion with 21
replac ed by 10 and 15, res pec tively .
Many of thos e c urious identities s tem from more general fac ts lik e the following: and Combining thes e gives us If n is an odd number (n = 2m + 1) we c an mak e us e of the s y mmetries to get The trans fer func tion of the Butterworth low pas s filter c an be ex pres s ed in terms of poly nomial and poles . By s etting the frequenc y as the c utoff frequenc y , the following identity c an be
proved: Computing π  An effic ient way to c ompute π is bas ed on the following identity without variables , due toMac hin: or, alternatively , by us ing an identity of Leonhard Euler: A useful mnemonic for certain values of sines and cosines
For c ertain s imple angles , the s ines and c os ines tak e the form Miscellany  for 0 ≤ n ≤ 4, whic h mak es them eas y to remember.  W ith the golden ratio φ: Als o s ee ex ac t trigonometric c ons tants . An identity of Euclid  Euc lid s howed in Book XIII, Propos ition 10 of his Elements that the area of the s quare on the s ide of a regular pentagon ins c ribed in a c irc le is equal to the s um of the areas of the s quares
on the s ides of the regular hex agon and the regular dec agon ins c ribed in the s ame c irc le. In the language of modern trigonometry , this s ay s : Ptolemy us ed this propos ition to c ompute s ome angles in his table of c hords . en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 10/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia Composition of trigonometric f unctions  This identity involves a trigonometric func tion of a trigonometric func tion: where J 0 and J 2k are Bes s el func tions . Calculus  In c...
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This document was uploaded on 02/04/2014.
- Spring '14