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This module is part of the collection, A First Course in Electrical and Computer Engineering . The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

e j θ , cos θ , and sin θ

e j θ = lim n ( 1 + j θ n ) n = n = 0 1 n ! ( j θ ) n = cos θ + j sin θ cos θ = n = 0 ( - 1 ) n ( 2 n ) ! θ 2 n ; sin θ = n = 0 ( - 1 ) n ( 2 n + 1 ) ! θ 2 n + 1

cos θ = n = 0 ( - 1 ) n ( 2 n ) ! θ 2 n ; sin θ = n = 0 ( - 1 ) n ( 2 n + 1 ) ! θ 2 n + 1

Trigonometric identities

sin 2 θ + cos 2 θ = 1
sin ( θ + φ ) = sin θ cos φ + cos θ sin φ
cos ( θ + φ ) = cos θ cos φ - sin θ sin φ
sin ( θ - φ ) = sin θ cos φ - cos θ sin φ
cos ( θ - φ ) = cos θ cos φ + sin θ sin φ

Euler's equations

e j θ = cos θ + j sin θ
sin θ = e j θ - e - j θ 2 j
cos θ = e j θ + e - j θ 2

De moivre's identity

( cos θ + j sin θ ) n = cos n θ + j sin n θ

Binomial expansion

( x + y ) N = n = 0 N N n x n y N - n ; N n = N ! ( N - n ) ! n !
2 N = n = 0 N N n

Geometric sums

k = 0 a z k = a 1 - z | z | < 1
k = 0 N - 1 a z k = a ( 1 - z N ) 1 - z z 1

Taylor's series

f ( x ) = k = 0 f ( k ) ( a ) ( x - a ) k k !
( Maclaurin's Series if a = 0 )

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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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