# Put vertical rotated text within minipage environment

Here's a tikz option that might suit. (Updated for US Letter size paper.)

\documentclass{article}
\usepackage[landscape,margin=0cm]{geometry}
\usepackage{mathtools}
\usepackage{amssymb,amsthm}
\usepackage{multicol}
\usepackage[most]{tcolorbox}
\usepackage{tikz}
\usetikzlibrary{positioning}

\newtcolorbox{mybox}[2][]{width=\linewidth,fontupper=\scriptsize,
fonttitle=\bfseries\sffamily\scriptsize, colbacktitle=black, enhanced,
attach boxed title to top left={yshift=-2mm,xshift=3mm},
boxed title style={sharp corners},top=2mm, bottom=1mm, left=1.5mm,
right=1.5mm, title=#2, colback=white}

\pagestyle{empty}
\parindent=0pt
\parskip=2pt

\begin{document}

\begin{multicols*}{3}

\begin{mybox}{Single Side-Band (SSB)}
\begin{tikzpicture}[node distance=2mm, inner sep=0pt, outer sep=0pt]
\node (image) {\includegraphics[width=2.7cm]{example-image-a}};
\node (phase) [align=center, text width=5.5cm, right=of image.north east,
anchor=north west] {
Phase Shifting method: \\
$\phi_{SSB} = A_{c}[m(t)\cos 2\pi f_{c}t\mp\hat{m}(t)\sin2\pi f_{c}t]$
};
\node [below=of image.south west, anchor=north west,text width=7.5cm] {
Demod of SSB-SC: \\
\quad$=A_{c}m(t)+\underbrace{A_{c}[m(t)\cos 4\pi f_{c}t\mp \hat{m}(t)\sin4\pi f_{c}t]}_{\text{Suppressed by LPF}}$};
\node [rotate=90, text width=8.1cm, below=of phase.south east, anchor=south
east, font=\tiny] {
\begin{align*}
\phi_{SSB} = & A_{c}[m(t)\cos 2\pi f_{c}t\mp\hat{m}(t)\sin2\pi f_{c}t] + A\cos2\pi f_{c}t, \\
= & E(t)\cos(2\pi f_{c}t + \theta(t)) \\
\text{where}\quad E(t) = & A_{c}\sqrt{[A+m(t)]^{2} + [\hat{m}(t)]^{2}} \text{ and } \theta(t)=-\tan^{-1}\left\{ \frac{\hat{m}(t)}{A + m(t)} \right\}
\end{align*}
};
\end{tikzpicture}
\end{mybox}

\end{multicols*}
\end{document}


Probably a year late for @Joe but I think is worth offering a xcoffin solution.

Many question posted in this forum are related with positioning graphics or text material in a page, according to a particular design: covers, title-pages, beamer slides, posters, TOC, etc.

I appreciate the package xcoffin because provides, within a very simple environment, a solution for most of my problems on those issues.

Over time I got rid of geometry, rotate, titlepage, positioning, float and similar packages.

xcoffin has only a handful of intuitive commands that allows the assembly of the boxes and the fine tuning quite fast. This is very important in the one-time jobs -- as I guess is this case --, like thesis or books covers, when you do not want to invest time in learning a new set of commands and the incompatibilities of a new package, to be used only once.

This particular problem is even easier than the example showed in the xcoffin documentation for two reason:

1. It does not have specific requirements over the relative position of the material with respect to another, thus allowing to use a single reference origin (in this case the insertion point) and

2. being a one time job (against to being a template or a class) there is not need to complicate the code with new lengths, commands or relative references to the boxes widths or heights, or different fonts.

The final tuning can be done manually, adjusting sizes and positions until a visually pleasant outcome is obtained.

Only five coffins are needed for the material supplied: the tcolorbox, the image-a and the three equations.

An extra box is used to collect the joined coffins and produce the output.

The proposed solution is independent of the page size and its orientation. (e.g. commenting out the geometry package).

One can only hope that xcoffin will some day abandon the x!

    \documentclass{article}

\usepackage[landscape,margin=0cm]{geometry}
\usepackage{xcolor}
\usepackage{amsmath}
\usepackage[most]{tcolorbox}
\usepackage{xcoffins}

%% from the original, not needed in this case. xcoffin will work in any case
%\usepackage{mathtools}
%\usepackage{multicol}
%\usepackage{amssymb,amsthm}
%\usepackage{tikz}
%\usepackage{rotating}
%\usepackage{float}

\pagestyle{empty}
\parindent=0pt
\parskip=2pt

\newlength{\lmargin}
\setlength{\lmargin}{1.5em}

\begin{document}

%%%%%%%%  Create & Set 6 coffins
\NewCoffin\CheatSheet
\NewCoffin\borderframe
\NewCoffin\imageA
\NewCoffin\eqPSM
\NewCoffin\eqDemod
\NewCoffin\eqPhi

\SetHorizontalCoffin\CheatSheet{} %Coffin to hold the combined joined output

\SetHorizontalCoffin\imageA{\includegraphics[width=2.7cm]{example-image-a}}

\SetHorizontalCoffin\eqDemod{%
Demod of SSB-SC: $=A_{c}m(t)+\underbrace{A_{c}[m(t)\cos 4\pi f_{c}t\mp \hat{m}(t)\sin4\pi f_{c}t]}_{\text{Suppressed by LPF}}$%
}

\SetVerticalCoffin\eqPSM{186pt}{%  PSM eq width =186pt
\centering Phase Shifting method:\\%
$\phi_{SSB} = A_{c}[m(t)\cos 2\pi f_{c}t\mp\hat{m}(t)\sin2\pi f_{c}t]$%
}

\SetVerticalCoffin\eqPhi{338pt}{% phi eq width =338pt
\begin{flalign*}
\phi_{SSB} &=  A_{c}[m(t)\cos 2\pi f_{c}t\mp\hat{m}(t)\sin2\pi f_{c}t] + A\cos2\pi f_{c}t,\\%
&=  E(t)\cos(2\pi f_{c}t + \theta(t))\\%
\text{where}\quad E(t) &=  A_{c}\sqrt{[A+m(t)]^{2} + [\hat{m}(t)]^{2}} \text{ and } \theta(t)=%
-\tan^{-1}\left\{ \frac{\hat{m}(t)}{A + m(t)} \right\}%
\end{flalign*}
}

\SetHorizontalCoffin\borderframe{%
\begin{tcolorbox}[%
width=338pt+3\lmargin,% phi eq width =338pt
height=338pt+2\lmargin,%  phi eq height =338pt after rotation
fontupper=\scriptsize,%
fonttitle=\bfseries\sffamily\scriptsize,%
colbacktitle=black,enhanced,%
attach boxed title to top left={yshift=-3mm,xshift=3mm},%
boxed title style={sharp corners},top=3pt,bottom=2pt,%
title=Single Side-Band (SSB),colback=white]
\end{tcolorbox}
}%

%%%%%%%  Assembly

\JoinCoffins*\CheatSheet[l,b]\borderframe[l,t]

\JoinCoffins*\CheatSheet[l,b]\imageA[l,t](\lmargin,  -2\baselineskip)

\JoinCoffins*\CheatSheet[l,b]\eqPSM[r,t](\lmargin+290pt, -2\baselineskip) %Demod eq width = 290pt

\JoinCoffins*\CheatSheet[l,b]\eqDemod[l,t](\lmargin, -3\baselineskip-58pt) % image A height =58pt

\RotateCoffin\eqPhi{-90}

\JoinCoffins*\CheatSheet[l,b]\eqPhi[l,b](\lmargin+290pt, -2\baselineskip)

%%%%%%%%%% Ouput

\noindent\TypesetCoffin\CheatSheet(20pt,-100pt) %20pt right & 100pt down from upper left insertion point

\end{document}