```
\documentclass{article}
\usepackage{tabularx}
\usepackage{natbib}
\bibpunct{(}{)}{;}{a}{,}{,} % I added this

\usepackage{verbatim}  % for commenting

\usepackage{float}
\usepackage{array}
\usepackage{multirow}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{cite}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{algorithmic}
\usepackage{graphicx}
\usepackage{textcomp}
\usepackage{xcolor}
\usepackage{verbatim}  % for commenting

\usepackage{tabularx}
\usepackage{booktabs}
\usepackage{caption}
\usepackage{amsmath}
\usepackage{siunitx}

\newcommand{\stone}{\theta_1}

\newcommand{\sx}{x}
\newcommand{\sy}{y}

\begin{document}

\begin{table} [tbp]
\centering
  \caption{Equations}
  \label{table:Equations}
  \begin{tabular}{p{0.45in} p{0.5in} p{2.4in}} % {@{}ll@{}}
    \toprule
    \textbf{Inputs} & \textbf{Output} & \textbf{Equation} \\
    \midrule

   \rotatebox[origin=c]{0}{$\stone$}
    & $\sx$ & \begin{equation}\tag{1}
        0.5 \cos(2\pi\stone) + 0.5
    \end{equation}  \\%\hline

   \rotatebox[origin=c]{0}{$\stone$}
    & $\sy$ & \begin{equation}\tag{2} 0.5\sin(2\pi\stone) + 0.5 
    \end{equation}\\%\hline

\bottomrule
\end{tabular}
\end{table}

\end{document}

```

This is how the table looks after the above code is executed:

[![enter image description here](https://i.stack.imgur.com/6rLO0.png)](https://i.stack.imgur.com/6rLO0.png)

In this, the columns of the table are not aligned properly. I need the equations to be labeled.