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Unformatted text preview: alc ulus the relations s tated below require angles to be meas ured in radians ; the relations would bec ome more c omplic ated if angles were meas ured in another unit s uc h as degrees . If the
trigonometric func tions are defined in terms of geometry , along with the definitions of arc length and area, their derivatives c an be found by verify ing two limits . The firs t is : verified us ing the unit c irc le and s queez e theorem. The s ec ond limit is : verified us ing the identity tan(x/2) = (1 − c os x)/s in x. Having es tablis hed thes e two limits , one c an us e the limit definition of the derivative and the addition theorems to s how that (s in x)
′ = c os x and (c os x)′ = − s in x. If the s ine and c os ine func tions are defined by their Tay lor s eries , then the derivatives c an be found by differentiating the power s eries termby term. The res t of the trigonometric func tions c an be differentiated us ing the above identities and the rules of differentiation:[37][38][39] The integral identities c an be found in "lis t of integrals of trigonometric func tions ". Some generic forms are lis ted below. Implications [edit] The fac t that the differentiation of trigonometric func tions (s ine and c os ine) res ults in linear c ombinations of the s ame two func tions is of fundamental importanc e to many fields of
mathematic s , inc luding differential equations and Fourier trans forms . Exponential def initions
Function [edit]
Inve rse function[40] en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 11/12 12/7/13 List of tr ig onometr ic identities  Wikipedia, the fr ee encyclopedia Miscellaneous [edit] Dirichlet kernel [edit] The Dirichle t ke rne l Dn(x) is the func tion oc c urring on both s ides of the nex t identity : The c onvolution of any integrable func tion of period 2π with the Diric hlet k ernel c oinc ides with the func tion's nthdegree Fourier approx imation. The s ame holds for any meas ure orgeneraliz ed
func tion. Tangent halfangle substitution [edit] Main artic le: Tangent halfangle s ub s titution
If we s et then[41] where ei x = c os (x) + i s in(x), s ometimes abbreviated to c is (x).
W hen this s ubs titution of t for tan(x/2) is us ed in c alc ulus , it follows that s in(x) is replac ed by 2t/(1 + t2), c os (x) is replac ed by (1 − t2)/(1 + t2) and the differential dx is replac ed by
(2 dt)/(1 + t2). Thereby one c onverts rational func tions of s in(x) and c os (x) to rational func tions of t in order to find their antiderivatives . See also [edit] en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 12/12...
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