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Unformatted text preview: a, Euler's formula and binomial theorem.
Cosine Product-to-sum and sum-to-product identities S i ne  The produc t-to-s um identities or pros thaphaeres is formulas c an be proven by ex panding their right-hand s ides us ing the angle addition theorems . See beat (ac ous tic s ) and phas e detec torfor
applic ations of the s um-to-produc t formulæ .
Product-to-sum  Sum -to-product en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 6/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia Other related identities  (Triple tangent identity ) In partic ular, the formula holds when x, y, and z are the three angles of any triangle.
(If any of x, y, z is a right angle, one s hould tak e both s ides to be ∞. This is neither + ∞ nor − ∞; for pres ent purpos es it mak es s ens e to add jus t one point at infinity to the real line,
that is approac hed by tan(θ) as tan(θ) either inc reas es through pos itive values or dec reas es through negative values . This is a one-point c ompac tific ation of the real line.)
(Triple c otangent identity ) Hermite's cotangent identity  Main artic le: Hermite's c otangent identity
Charles Hermite demons trated the following identity . Suppos e a1, ..., an are c omplex numbers , no two of whic h differ by an integer multiple of π. Let (in partic ular, A1,1, being an empty produc t, is 1). Then The s imples t non-trivial ex ample is the c as e n = 2: Ptolemy's theorem  (The firs t three equalities are trivial; the fourth is the s ubs tanc e of this identity .) Es s entially this is Ptolemy 's theorem adapted to the language of modern trigonometry . Linear combinations  For s ome purpos es it is important to k now that any linear c ombination of s ine waves of the s ame period or frequenc y but different phas e s hifts is als o a s ine wave with the s ame period or
frequenc y , but a different phas e s hift. In the c as e of a non-z ero linear c ombination of a s ine and c os ine wave (whic h is jus t a s ine wave with a phas e s hift of π/2), we have where or equivalently or even or us ing the atan2 func tion More generally , for an arbitrary phas e s hift, we have where and The general c as e r...
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- Spring '14