# 12: Useful Mathematical Identities

## ejθ, cos θ, and sin θ

$e^{jθ} = \lim_{n\to \infty} \Bigl(1 + j\frac {θ} {n}\Bigr)^n = \sum_{n=0}^{\infty} {\frac {1} {n!}} (jθ)^n = \cos θ +j\sin θ$

$\cos θ = \sum_{n=0}^{\infty} \frac {(-1)^n} {(2n)!} θ^{2n}$

$\sin θ = \sum_{n=0}^{\infty} \frac {(-1)^n} {(2n+1)!} θ^{2n+1}$

## 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

$e^{jθ} = \cos θ + j\sin θ$

$\sin θ = \frac {e^{jθ}-e^{-jθ}} {2j}$

$\cos θ = \frac {e^{jθ}+e^{-jθ}} {2}$

## De Moivre's Identity

$(\cos θ +j\sin θ)^n = \cos {nθ} + j\sin {nθ}$

## Binomial Expansion

$(x+y)^N = \sum_{n=0}^{N} {\Bigl(\frac {N!} {(N-n)!n!} \Bigr)x^n y^{N-n}}$

$2^N = \sum_{n=0}^{N} {\Bigl(\frac {N!} {(N-n)!n!} \Bigr)}$

## Geometric Sums

$\sum_{k=0}^{\infty} {az^k} = \frac {a} {1-z} ;|z| \lt 1$

$\sum_{k=0}^{N-1} {az^k} = \frac {a\left(1-z^N\right)} {1-z} ;z \neq 1$

## Taylor's Series

$f(x) = \sum_{k=0}^{\infty} {f^{(k)}(a)\frac {(x-a)^k} {k!}}$

(Maclaurin's Series if a=0)