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Unformatted text preview: s ult as the rotation by the s um of the angles . Sines and cosines of sums of infinitely many terms  In thes e two identities an as y mmetry appears that is not s een in the c as e of s ums of finitely many terms : in eac h produc t, there are only finitely many s ine fac tors and c ofinitely many
c os ine fac tors .
If only finitely many of the terms θi are nonz ero, then only finitely many of the terms on the right s ide will be nonz ero bec aus e s ine fac tors will vanis h, and in eac h term, all but finitely many
of the c os ine fac tors will be unity . Tangents of sums  en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 3/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia Let ek (for k = 0, 1, 2, 3, ...) be the k th-degree elementary s y mmetric poly nomial in the variables
for i = 0, 1, 2, 3, ..., i.e., Then The number of terms on the right s ide depends on the number of terms on the left s ide.
For ex ample: and s o on. The c as e of only finitely many terms c an be proved by mathematic al induc tion. Secants and cosecants of sums  where ek is the k th-degree elementary s y mmetric poly nomial in the n variables xi = tan θi ,i = 1, ..., n, and the number of terms in the denominator and the number of fac tors in the produc t in
the numerator depend on the number of terms in the s um on the left. The c as e of only finitely many terms c an be proved by mathematic al induc tion on the number of s uc h terms . The
c onvergenc e of the s eries in the denominators c an be s hown by writing the s ec ant identity in the form and then obs erving that the left s ide c onverges if the right s ide c onverges , and s imilarly for the c os ec ant identity .
For ex ample, Multiple-angle f ormulae 
 Tn is the nth Che byshe v polynom ia l
Sn is the nth spre a d polynom ia l
de Moivre 's form ula ,  is the im a gina ry unit Double-angle, triple-angle, and half-angle formulae  See als o: Tangent half-angle formula
Thes e c an be s hown by us ing either the s um and differenc e identities or the multiple-angle formulae.
Double -a ngle form ula e  en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 4/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia Triple -a ngle form ula e  Ha lf-a ngle form ula e  The fac t that the triple-angle formula for s ine and c os ine only involves powers of a s ingle func tion allows one to relate the geometric problem of a c ompas s and s traightedge
c ons truc tion of angle tris ec tion to the algebraic problem of s olving a c ubic equation, whic h allows one to prove that this...
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This document was uploaded on 02/04/2014.
- Spring '14