129 lines
4.5 KiB
TeX
129 lines
4.5 KiB
TeX
\documentclass[a5paper,8pt]{extarticle}
|
|
|
|
\usepackage[margin=0.5cm]{geometry}
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage{amsfonts}
|
|
\usepackage{amsmath}
|
|
%\usepackage{gensymb}
|
|
\usepackage{polski}
|
|
\usepackage{multirow}
|
|
\usepackage{multicol}
|
|
|
|
\setlength{\multicolsep}{0pt}
|
|
|
|
\usepackage{titlesec}
|
|
\titleformat{\section} {\normalfont\Large\bfseries}{}{0em}{}
|
|
\titleformat{\subsection}{\normalfont\large\bfseries}{}{0em}{}
|
|
|
|
\title{\textbf{Algorytmy kwantowe}: trygonometria i liczby zespolone}
|
|
\date{2021-02-27}
|
|
\author{Robert Bendun (\texttt{robert@bendun.cc})}
|
|
|
|
\newcommand{\mi}{{i\mkern1mu}}
|
|
\renewcommand{\arraystretch}{1.5}
|
|
|
|
\begin{document}
|
|
|
|
\begin{center}
|
|
\makeatletter
|
|
{\Large \@title} \\ \@date, \@author \\
|
|
\makeatother
|
|
\end{center}
|
|
|
|
\section{Trygonometria}
|
|
|
|
\subsection{Wartości}
|
|
|
|
\begin{center}
|
|
\begin{tabular}{ |c|c|c|c|c|c| } \hline
|
|
$\alpha$ (deg) & $0\textdegree$ & $30\textdegree$ & $45\textdegree$ & $60\textdegree$ & $90\textdegree$ \\ \hline
|
|
$\alpha$ (rad) & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ \\ \hline
|
|
$\sin$ & $0$ & $\frac{1}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1$ \\ \hline
|
|
$\cos$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2}$ & $0$ \\ \hline
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
\subsection{Wzory redukcyjne}
|
|
|
|
\begin{center}
|
|
\begin{tabular}{ c | c | c | c }
|
|
\multicolumn{2}{c|}{$ \sin -\alpha = -\sin \alpha $} &
|
|
\multicolumn{2}{c}{$ \cos -\alpha = \sin \alpha $} \\ \hline
|
|
|
|
$ \sin \left( \frac{\pi}{2} - \alpha \right) = \cos \alpha $ &
|
|
$ \sin \left( \frac{\pi}{2} + \alpha \right) = \cos \alpha $ &
|
|
$ \cos \left( \frac{\pi}{2} - \alpha \right) = \sin \alpha $ &
|
|
$ \cos \left( \frac{\pi}{2} + \alpha \right) = -\sin \alpha $ \\
|
|
|
|
$ \sin \left( \pi - \alpha \right) = \sin \alpha $ &
|
|
$ \sin \left( \pi + \alpha \right) = -\sin \alpha $ &
|
|
$ \cos \left( \pi - \alpha \right) = -\cos \alpha $ &
|
|
$ \cos \left( \pi + \alpha \right) = -\cos \alpha $ \\
|
|
|
|
\hline
|
|
|
|
$ \sin \left( \frac{3\pi}{2} - \alpha \right) = -\cos \alpha $ &
|
|
$ \sin \left( \frac{3\pi}{2} + \alpha \right) = -\cos \alpha $ &
|
|
$ \cos \left( \frac{3\pi}{2} - \alpha \right) = -\sin \alpha $ &
|
|
$ \cos \left( \frac{3\pi}{2} + \alpha \right) = \sin \alpha $ \\
|
|
|
|
$ \sin \left( 2\pi - \alpha \right) = -\sin \alpha $ &
|
|
$ \sin \left( 2\pi + \alpha \right) = \sin \alpha $ &
|
|
$ \cos \left( 2\pi - \alpha \right) = \cos \alpha $ &
|
|
$ \cos \left( 2\pi + \alpha \right) = \cos \alpha $
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
\subsection{Tożsamości}
|
|
|
|
\begin{multicols}{2}
|
|
\begin{description}
|
|
\item $ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta $
|
|
\item $ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta $
|
|
\item $ \sin2\alpha = 2\sin\alpha\cos\alpha $
|
|
\item $ \cos2\alpha = 2\cos^2\alpha - 1 $
|
|
\end{description}
|
|
\end{multicols}
|
|
|
|
\section{Liczby zespolone}
|
|
|
|
\subsection{Postać algebraiczna}
|
|
|
|
\begin{multicols}{2}
|
|
\begin{description}
|
|
\item $ \alpha \pm \beta = \left( a + b\mi \right) \pm \left( c + d\mi \right)
|
|
= \left( a \pm c \right) + \left( b \pm d \right)\mi$
|
|
\item $ \alpha\beta = \left( a + b\mi \right) \left( c + d\mi \right)
|
|
= \left( ac - bd \right) + \left( bc + ad \right)\mi$
|
|
\item $ \frac{\alpha}{\beta} = \frac{a + b\mi}{c + d\mi} = \frac{(ac + bd) + (bc - ad)\mi}{c^2 + d^2} $
|
|
\item[Norma] $ |\alpha| = |a + b\mi| = \sqrt{a^2 + b^2} $
|
|
\item[Sprzężenie] $ \overline{a + \mi b} = a - b\mi $
|
|
\item $ \alpha\overline{\alpha} = (a + b\mi)(a - b\mi) = a^2 + b^2 = |\alpha|^2 $
|
|
\end{description}
|
|
\end{multicols}
|
|
|
|
\subsection{Postać trygonometryczna}
|
|
|
|
$ z = |z|\left( \frac{a}{|z|} + \frac{b}{|z|}\mi \right) $ ponieważ $ \sin\rho = \frac{b}{|z|} $ i $ \cos\rho = \frac{a}{|z|} $ mamy równość:
|
|
|
|
$$ z = a + b\mi = |z|(\cos\rho + \mi\sin\rho) $$
|
|
|
|
\begin{description}
|
|
\item $ xy = |x|(\cos \alpha + \mi\sin\alpha) \times |y|(\cos \beta + \mi\sin\beta) =
|
|
|x||y|\left[\cos(\alpha + \beta) + \mi\sin(\alpha+\beta)\right]$
|
|
|
|
\item $ \frac{x}{y} = |x|(\cos \alpha + \mi\sin\alpha) \div |y|(\cos \beta + \mi\sin\beta) =
|
|
\frac{|x|}{|y|}\left[\cos(\alpha - \beta) + \mi\sin(\alpha-\beta)\right]$
|
|
|
|
\item[Wzór de Moivre'a] $ z^n = |z|^n\left(\cos(n\rho) + \mi\sin(n\rho)\right) $
|
|
|
|
\item[Pierwiastki] $ \sqrt[n]{ z } = \left\{ \sqrt[n]{|z|} \left(\cos \frac{\rho + 2k\pi}{n} + \mi\sin \frac{\rho + 2k\pi}{n} \right) \mid k = 0, 1, 2, ..., n-1 \right\} $
|
|
|
|
\item[Wzór Eulera] $ e^{\theta\mi} = \cos\theta + \mi\sin\theta $
|
|
|
|
\end{description}
|
|
|
|
$$ \sin\theta = \frac{e^{\mi\theta} - e^{-\mi\theta}}{2\mi} \quad\quad\quad
|
|
\cos\theta = \frac{e^{\mi\theta} + e^{-\mi\theta}}{2\mi} $$
|
|
\end{document}
|