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\title{Algunas Fórmulas Útiles}
\author{Análisis Numérico -- Segundo Cuatrimestre de 2016}
\date{}

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\begin{abstract}
Por favor tome todas las medidas necesarias para asegurarse de que la información provista a continuación
es correcta y ha sido verificada, y utilícela en todo caso bajo su exclusiva responsabilidad.
\end{abstract}


\section*{Identidades trigonométricas}

\begin{enumerate}[i.]
\item $1 = \sin^2(\theta) + \cos^2(\theta)$
\item $1-\cos(\theta) = 2\sin^2(\theta/2)$
\item $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
\item $\cos(2\theta) = \sin^2(\theta)\cos^2(\theta)$
\item $e^{i\theta} + e^{-i\theta} = 2\cos(\theta)$
\item $e^{i\theta} - e^{-i\theta} = 2i\sin(\theta)$
\item $e^{i\theta} - 2 + e^{-i\theta} = -4\sin^2(\theta/2)$
\item  \[
\begin{array}{rl}
 z & = 1 - 4\alpha\sin^2(\theta/2) - 2 i \alpha \beta \sin(\theta) \qquad \mbox{entonces, por (iii) y (i),} \\[0.25cm]
 |z|^2 & = 1 - 8\alpha\sin^2(\theta/2) \left[ (1 - 2\alpha) \sin^2(\theta/2) + (1-2\alpha \beta^2) \cos^2(\theta/2) \right]
\end{array}
\]
\end{enumerate}

\section*{Expansiones de Taylor}

\begin{enumerate}[i.]
 \item $\sin(\varepsilon) \sim \varepsilon - \frac{\varepsilon^3}{6} + O(\varepsilon^5)$
 \item $\cos(\varepsilon) \sim 1 - \frac{\varepsilon^2}{2} + \frac{\varepsilon^4}{24} + O(\varepsilon^6)$
 \item $\sin^2(\varepsilon) \sim \varepsilon^2 - \frac{\varepsilon^4}{3} + O(\varepsilon^6)$
 \item $\cos^2(\varepsilon) \sim 1 - \varepsilon^2 + \frac{\varepsilon^4}{3} + O(\varepsilon^6)$
 \item $\sqrt{1-\alpha \sin ^2(\varepsilon)} \sim 1-\frac{\alpha \varepsilon^2}{2}+\left(\frac{\alpha}{6}-\frac{\alpha^2}{8}\right) \varepsilon^4+O\left(\varepsilon^6\right) $
 \item $\tan^{-1}\left(\frac{\alpha \sin (\varepsilon)}{1-\beta \sin ^2\left(\frac{\varepsilon}{2}\right)}\right) \sim \alpha \varepsilon+\frac{1}{12} \left(-4 \alpha^3+3 \alpha \beta-2 \alpha\right)\varepsilon^3 + O\left(\varepsilon^5\right)$
 \item $\tan ^{-1}\left(\frac{\alpha \sin (\varepsilon)}{\sqrt{1-\beta \sin ^2(\varepsilon)}}\right) \sim \alpha \varepsilon+\frac{1}{6} \left(-2 \alpha^3+3 \alpha \beta-\alpha\right)\varepsilon^3 + O\left(\varepsilon^5\right)$
 \item $\tan ^{-1}\left(\frac{\alpha \sin (\varepsilon)}{\alpha \cos (\varepsilon)+(1-\alpha)}\right) \sim \alpha \varepsilon+\frac{1}{6} \left(-2 \alpha^3+3 \alpha^2-\alpha\right) \varepsilon^3+O\left(\varepsilon^5\right)$
\end{enumerate}




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