Merge pull request #5 from Electrostat-Lab/latex-support

Latex support

.github/workflows/jekyll-gh-pages.yml

@@ -28,6 +28,10 @@ jobs:
     steps:
       - name: Checkout
         uses: actions/checkout@v4
+	  - name: Install TEX-Live Base and Latex2html
+		run: sudo apt install texlive-base latex2html
+	  - name: Build LaTEX Code
+		run: cd ./Math-I-LaTEX && chmod +rwx ./build-script/compile-tex math-i.tex && ./build-script/compile-tex math-i.tex
       - name: Setup Pages
         uses: actions/configure-pages@v5
       - name: Build with Jekyll

.gitignore

@@ -0,0 +1 @@
+*/math-i/

Math-I-LaTEX/build-script/compile-tex.sh

@@ -0,0 +1,6 @@
+#!/bin/bash
+
+input="${1}"
+
+pdflatex $input
+latex2html $input

Math-I-LaTEX/lib/text.tex

@@ -0,0 +1,8 @@
+% Define custom subsection command with color parameter
+\newcommand{\specialsubsection}[3]{
+    \medskip
+    \begin{tcolorbox}[colback=#3!10!white, colframe=#3!80!black, title=\textbf{#1}, sharp corners=south]
+        #2
+    \end{tcolorbox}
+    \medskip
+}

Math-I-LaTEX/math-i.tex

@@ -0,0 +1,27 @@
+\documentclass[0pt, a4paper]{article}
+\usepackage{amsmath} % For advanced math typesetting
+\usepackage{authblk} % Load authblk for multiple authors and affiliations
+\usepackage{xcolor}
+\usepackage{tcolorbox} %LaTeX package to custom color boxes for custom subsectioning
+\usepackage{graphicx} %LaTeX package to import graphics
+\usepackage{amsthm}
+\usepackage{amssymb} %LaTeX package to import extra symbols
+\usepackage{cancel} %LaTeX package to import cancelling
+
+\graphicspath{{assets/}} %configuring the graphicx package
+
+\begin{document}
+
+\title{The Electrostatic SDK Linear Projection Project: Mathematics-I}
+
+\author{Project Lead: Pavly Gerges}
+\affil[0]{Part of the Electrostatic-Sandbox SDK, Project: ElectroNetSoft.
+A generalized Math Framework for Linear Projection Algorithms, purely written in ISO/C99.}
+\date{\today}
+
+\maketitle
+
+\input{trigonometry/trig}
+
+\end{document}
+

Math-I-LaTEX/trigonometry/analytical.tex

@@ -0,0 +1,24 @@
+\input{lib/text}
+
+\subsubsection{Fundamentals Trigonometric identities}
+
+\input{trigonometry/analytical/fundamentals}
+
+\subsubsection{Co-functions Formulas}
+
+\input{trigonometry/analytical/co-functions}
+
+\subsubsection{The Addition Formulas}
+
+\input{trigonometry/analytical/addition-formulas}
+
+\subsubsection{Co-functions Addition Formulas}
+
+\subsubsection{Double Angle Formulas}
+
+\subsubsection{Half-Angle Formulas}
+
+\subsubsection{Product-Sum Formulas}
+
+\subsubsection{Miscellaneous Equations}
+

Math-I-LaTEX/trigonometry/analytical/addition-formulas.tex

@@ -0,0 +1,224 @@
+The following formulas are of the most essential trigonometric formulas that 
+will be utilized in the subsequent text, their proof is attached by the end of this section: 
+
+\begin{itemize}
+    \item \( \cos{(\phi \pm \alpha)} = \cos{(\phi) \cdot \cos{(\alpha)}} 
+            \mp \sin{(\phi)} \cdot \sin{(\alpha)} \)    <<1>> 
+    \item \( \sin{(\phi \pm \alpha)} = \sin{(\phi)} \cdot \cos{(\alpha)} 
+            \pm \cos{(\phi)} \cdot \sin{(\alpha)} \)    <<2>> 
+    \item \( \tan{(\phi \pm \alpha)} = \frac{\tan{\phi} \pm \tan{\alpha}}{1 \mp \tan{\phi} \cdot \tan{\alpha}} \)  <<3>>
+
+\end{itemize}
+
+\specialsubsection{Proof for equation (1) using Vector Maths from Calculus II:}{
+\includegraphics{proof-1.png} 
+\\\\ 
+\(   
+\because
+\vec{\Phi} = \vec{\Phi} - \vec{\rho} 
+           = (\vec{{\Phi}_x} + \vec{{\Phi}_y}) - \vec{\rho}
+            = (\begin{bmatrix}
+            \cos{\phi} \\
+            0
+            \end{bmatrix} + \begin{bmatrix}
+            0 \\
+            \sin{\phi}
+            \end{bmatrix}) - \begin{bmatrix}
+            0 \\
+            0
+            \end{bmatrix} = \begin{bmatrix}
+                            \cos{\phi} \\
+                            \sin{\phi}
+                            \end{bmatrix} \)
+
+\( 
+\vec{A} = \vec{A} - \vec{\rho}
+             = (\vec{{A}_x} + \vec{{A}_y}) - \vec{\rho}
+             = \begin{bmatrix}
+                \cos{\alpha} \\
+                \sin{\alpha}
+                \end{bmatrix}
+\)
+
+\(
+\vec{\psi} = \vec{\psi} - \vec{\rho}
+           = (\vec{{\psi}_x} + \vec{{\psi}_y}) - \vec{{A}_x}
+           = \vec{\upsilon} + \vec{\Pi}
+\)
+
+\(
+\vec{\upsilon} = \vec{\upsilon} - \vec{\rho} 
+               = \vec{A} - \vec{A_x}
+               = \begin{bmatrix}
+                0 \\
+                \sin{\alpha}
+                \end{bmatrix}
+\)
+
+\begin{equation}
+\therefore \vec{\Pi} = \vec{\Pi} - \vec{\rho} 
+          =  \begin{bmatrix}
+            \cos{\beta} \\
+            \sin{\beta}
+            \end{bmatrix} - \vec{\rho}    
+            \\
+          = \vec{\Phi} - \vec{A} 
+          =  \begin{bmatrix} 
+            \cos{\phi} - \cos{\alpha} \\
+            \sin{\phi} - \sin{\alpha}
+            \end{bmatrix}
+\end{equation}
+
+
+Essentially, in order to compute for $\angle{\beta}$ in an $R^2$ vectorspace, the initial side angle 
+must align with the \(+'ve\) direction of the x-axis. 
+\\\\
+\(\because \vec{\Pi}\) is translated by vector \(\vec{A}\)
+of polar coordinates \([\cos{\alpha}, \sin{\alpha}]\) then to
+align \(\vec{\Pi}\) to the \(+'ve\) direction of the x-axis, one extra
+operation should be attained, a vector translation transformation that 
+could be carried on through the preset vector \( \vec{\rho} \):
+\\
+\(
+\vec{\Pi} = \vec{\Pi} - \vec{\rho} \\
+          = (\vec{A} + \vec{\Pi}) - \vec{A} \\
+          = \begin{bmatrix}
+            (\cos{\alpha} + \cos{\beta}) - \cos{\alpha}\\
+            (\sin{\alpha} + \sin{\beta}) - \sin{\alpha}
+            \end{bmatrix} \\
+          = \begin{bmatrix}
+            \cos{\beta} \\
+            \sin{\beta}
+            \end{bmatrix} \\
+\)
+\\ 
+After this transformation, vector \(\vec{\Pi}\) aligns perfectly with the 
+\(+'ve\) direction of x-axis rendering computing angle \(\angle{\beta}\) straightforward,
+and hence the vector \(\vec{\Pi}\) can be calculated using distance operations: 
+\\ 
+\(
+\vec{\Pi} = \vec{\Pi} - \vec{\rho} \\
+          = \vec{\Pi} - \vec{x} \\
+          = \begin{bmatrix}
+            \cos{\beta} - 1\\
+            \sin{\beta} - 0
+            \end{bmatrix} \\
+\)
+\\
+\begin{equation}
+\because 
+\angle{\phi} = \angle{\alpha} + \angle{\beta} \\
+\\
+\therefore \angle{\beta} = \angle{\phi} - \angle{\alpha} \\
+\\
+\therefore \vec{\Pi} = \begin{bmatrix}
+    \cos{(\phi - \alpha)} - 1\\
+    \sin{(\phi - \alpha)}
+    \end{bmatrix} \\
+\end{equation}
+\\
+\( \therefore, from (1) and (2), \) we can deduce that the following norms are equal:
+\\
+\(
+\sqrt{{(\cos{\phi} - \cos{\alpha})}^2 + {(\sin{\phi} - \sin{\alpha})}^2} \\
+= \sqrt{{(\cos{\phi - \alpha} - 1)}^2 + {(\sin{\phi - \alpha})}^2} 
+\\ \cdots \\
+2 \cdot (1 - \cos{(\phi - \alpha)}) = 2 \cdot (1 - \cos{\phi}\cdot\cos{\alpha} - \sin{\phi}\cdot\sin{\alpha})
+\\ \cdots \\
+\)
+\specialsubsection{The Cosine Addition Formula}{
+\[
+\therefore \cos{(\phi - \alpha)} = \cos{\phi}\cdot\cos{\alpha} + \sin{\phi}\cdot\sin{\alpha}
+\]
+}{red}
+
+}{black}
+
+\specialsubsection{Proof for equation (2) using the Co-Function forumlas}{
+\( \because \sin{\theta} = \cos{(\frac{\pi}{2} - \theta)} \); as both cosine and sine functions are co-functions
+in the same right-angled triangle or among 2 similar right-angled triangles in a unit circle.
+\\\\
+\(
+\sin{(\phi \pm \alpha)} = \cos{(\frac{\pi}{2} - (\phi \pm \alpha))} 
+                      = \cos{((\frac{\pi}{2} - \phi) \mp \alpha)} \\
+                      = \cos{(\frac{\pi}{2} - \phi)}\cdot\cos{\alpha} \pm \sin{(\frac{\pi}{2} - \phi)}\cdot\sin{\alpha} \\
+                      = \sin{\phi}\cdot\cos{\alpha} \pm \cos{\phi}\cdot\sin{\alpha}
+\)
+\specialsubsection{The Sine Addition Formula}{
+\[
+\therefore \sin{(\phi \pm \alpha)} = \sin{\phi}\cdot\cos{\alpha} \pm \cos{\phi}\cdot\sin{\alpha}
+\]
+}{red}
+}{black}
+
+\specialsubsection{Proof for equation (3) using the Tangent fundamental identities}{
+\( \because \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \), and from (1) and (2):
+\\
+\begin{equation} 
+\tan{(\phi \pm \alpha)} = \frac{\sin{(\phi \pm \alpha)}}{\cos{(\phi \pm \alpha)}} \\
+             = \frac{
+                \sin{\phi}\cdot\cos{\alpha} \pm \cos{\phi}\cdot\sin{\alpha}
+             }{
+                \cos{\phi}\cdot\cos{\alpha} \mp \sin{\phi}\cdot\sin{\alpha}
+             }
+\end{equation}
+\\ 
+\( \because \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \\
+    \therefore \sin{\theta} = \cos{\theta}\cdot\tan{\theta} 
+\)
+\\
+\(  
+\therefore \tan{(\phi \pm \alpha)} =  \frac{
+    \tan{\phi}\cdot\cancel{\cos{\phi}\cdot\cos{\alpha}} \pm \cancel{\cos{\phi}\cdot\cos{\alpha}}\cdot\tan{\alpha}
+ }{
+    \cancel{\cos{\phi}\cdot\cos{\alpha}} \mp \cancel{\cos{\phi}\cdot\cos{\alpha}}\cdot\tan{\alpha}\cdot\tan{\phi}
+ }
+\)
+\\
+\specialsubsection{The Tangent Addition Formula}{
+\[
+    \therefore \tan{(\phi \pm \alpha)} = \frac{
+        \tan{\phi} \pm \tan{\alpha}
+     }{
+        1 \mp \tan{\alpha}\cdot\tan{\phi}
+     }
+\]
+}{red}
+
+}{black}
+
+\specialsubsection{Extra Verification Proofs done differently}{
+\specialsubsection{
+Verify that: \(
+\sin{(\phi + \frac{3\pi}{2})} = - \cos{\phi}
+\)
+\\
+}{
+
+\begin{proof}
+\begin{equation} 
+\because \sin{(\phi + \frac{3\pi}{2})} = \sin{(\phi + \pi + \frac{\pi}{2})}
+                                       = \sin{(\frac{\pi}{2} - (-\phi - \pi))}
+\end{equation}
+
+\begin{equation} 
+\because \sin{(\frac{\pi}{2} - \alpha)} = \cos{\alpha} 
+\end{equation}
+
+\begin{equation}
+         \cos{(-\alpha)} = \cos{\alpha}
+\end{equation} 
+
+
+\( \therefore \) from (4), (5), \& (6):
+\\
+\(
+\sin{(\phi + \frac{3\pi}{2})} = \sin{(\frac{\pi}{2} - (-\phi - \pi))} \\
+                              = \cos{(-\phi - \pi)} \\
+                              = \cos{(-(\phi + \pi))} \\
+                              = \cos{(\phi + \pi)} \\
+                              = - \cos(\phi)
+\)
+\end{proof}
+}{red}
+}{black}

Math-I-LaTEX/trigonometry/analytical/co-functions.tex

@@ -0,0 +1,49 @@
+This section concludes proofs and formulas for the Sine and Cosine interaction as Co-functions.
+\\
+
+\specialsubsection{Co-function Formulas}{
+The following is the summary of the re-usable formulas, notice how these will be utilize dramatically
+in the subsequent sections, if \(\angle{\theta}\) is an acute angle, then the following holds true:
+\\
+\begin{equation}
+    \cos{\theta} = \sin{(\frac{\pi}{2} - \theta)}
+    \label{eq:equation}
+\end{equation}
+
+\begin{equation}
+    \sin{\theta} = \cos{(\frac{\pi}{2} - \theta)}
+    \label{eq:equation2}
+\end{equation}
+
+\begin{equation}
+    \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \\
+                 = \frac{\cos{(\frac{\pi}{2} - \theta})}{\sin{(\frac{\pi}{2} - \theta})} \\
+    \label{eq:equation3}
+\end{equation}
+
+\begin{equation}
+    \sin{(\theta + \frac{\pi}{2})} = \cos{(\frac{\pi}{2} - (\theta + \frac{\pi}{2}))} \\
+                                 = \cos{(-\theta)} = \cos{\theta}
+    \label{eq:equation4}
+\end{equation}
+
+\(\because \text{\(\angle{(\theta + \frac{\pi}{2})}\) and \(\angle{\alpha}\) are supplementary angles (i.e., \(\angle{(\theta + \frac{\pi}{2})} +  \angle{\alpha} = \angle{\pi}).\)}\\
+    \therefore \sin{(\theta + \frac{\pi}{2})} = \sin{(\pi - \alpha)} \\
+                                              = \sin{(\pi - \alpha)} \\
+\)
+
+\begin{equation}
+    \therefore \sin{(\theta + \frac{\pi}{2})} = \sin{(\pi - \alpha)}
+    \label{eq:equation5}
+\end{equation}
+
+\begin{equation}
+    \sin{(\theta + \pi)} = \sin{(3\frac{\pi}{2} - \alpha)}
+    \label{eq:equation6}
+\end{equation}
+
+}{red}
+
+\specialsubsection{Proof for the co-functions}{
+
+}{black}

Math-I-LaTEX/trigonometry/analytical/fundamentals.tex

@@ -0,0 +1,30 @@
+This section is devoted to the deduction of the fundamental trigonometric identities 
+from the unit circle, and the Pythagorean theorem.
+
+First off, here is an exhaustive list of the fundamental trigonometric identities before heading 
+to their deduction, although not commonly encountered, but it's useful in particular 
+scenarios in software design:
+
+If angle \( \angle{\theta} \) is a central angle inscribed by a unit circle \( C \),
+with its initial side coincident with the positive direction of the x-axis, and the 
+terminal side lies in any of the 4 quadrants in an \( R^2 \) vector-space, then the following holds:
+
+\begin{itemize}
+    \item \( \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \)
+    \item \( \cot{\theta} = \frac{1}{\tan{\theta}} = \frac{\cos{\theta}}{\sin{\theta}} \)
+    \item \( \csc{\theta} = \frac{1}{\sin{\theta}} \)
+    \item \( \sec{\theta} = \frac{1}{\cos{\theta}} \)
+    \item \( \sin^2{\theta} + \cos^2{\theta} = r^2 = 1 \)
+    \item \( \tan^2{\theta} + 1 = \sec^2{\theta} \)
+    \item \( 1 + \cot^2{\theta} = \csc^2{\theta} \)
+\end{itemize}
+
+\specialsubsection{Alternative forms for the fundamental identities:}{
+\begin{itemize}
+    \item \( \tan{\theta} = \frac{\sec{\theta}}{\csc{\theta}} \)
+    \item \( \sin^2{\theta} = 1 - \cos^2{\theta} \)
+    \item \( \cos^2{\theta} = 1 - \sin^2{\theta} \)
+    \item \( \tan^2{\theta} = \sec^2{\theta} - 1 \)
+    \item \( \cot^2{\theta} = \csc^2{\theta} - 1 \)
+\end{itemize}
+}{black}