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Engineering LibreTexts

12: Useful Mathematical Identities

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  • e, 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)