\documentclass{article}
\usepackage{graphicx} % Required for inserting images
\usepackage{amsmath}

\title{Basic Trig}
\author{lemtrees }
\date{September 2023}

\begin{document}

\maketitle

\section{Variables and Definitions}

\begin{align*}
H & : \text{Height of the satellite above Earth's surface (in km)} \\
L & : \text{Length of the airplane (in feet)} \\
A & : \text{Altitude of the airplane above sea level (in feet)} \\
P & : \text{Percent larger the plane appears to the satellite} \\
\theta_0 & : \text{Angle at the satellite when the plane is at sea level} \\
D_0 & : \text{Hypotenuse of the triangle when the plane is at sea level (in feet)} \\
\theta_A & : \text{Angle at the satellite when the plane is at altitude \( A \)} \\
D_A & : \text{Hypotenuse of the triangle when the plane is at altitude \( A \) (in feet)}
\end{align*}

\newpage

\section{Figure of Plane at Sea Level}

\begin{figure}[h]
    \centering
    \includegraphics[width=1\linewidth]{planeatGround.png}
    \caption{Plane at Sea Level}
    \label{fig:enter-label1}

    \includegraphics[width=1\linewidth]{planeatA.png}
    \caption{Plane at altitude A}
    \label{fig:enter-label2}
\end{figure}

\section{Equations for Angles}

The angle \( \theta_0 \) at the satellite when the plane is at sea level is defined as:

\begin{equation}
\theta_0 = \tan^{-1}\left(\frac{L}{H}\right)
\end{equation}

When the plane is at an altitude \( A \), the angle \( \theta_A \) at the satellite is:

\begin{equation}
\theta_A = \tan^{-1}\left(\frac{L}{H - A}\right)
\end{equation}


\section{Calculating the Percent Increase in Apparent Size}

The percent larger \( P \) that the plane would appear to be is calculated as:

\begin{equation}
P = \frac{\theta_A - \theta_0}{\theta_0} \times 100
\end{equation}

The expanded equation for \( P \) in terms of \( H \), \( L \), and \( A \) is:

\begin{equation}
P = \frac{\tan^{-1}\left(\frac{L}{H - A}\right) - \tan^{-1}\left(\frac{L}{H}\right)}{\tan^{-1}\left(\frac{L}{H}\right)} \times 100
\end{equation}


To account for unit conversions, where \( H \) is in kilometers, and \( A \) and \( L \) are in feet, the equation for \( P \) becomes:

\begin{equation}
P = \frac{\tan^{-1}\left(\frac{L}{H \times 3280.84 - A}\right) - \tan^{-1}\left(\frac{L}{H \times 3280.84}\right)}{\tan^{-1}\left(\frac{L}{H \times 3280.84}\right)} \times 100
\end{equation}


\section{Calculation with Specific Values}

The equation for \( P \) with \( A = 35000 \) feet, \( L = 209 \) feet, and \( H = 700 \) km is:

\begin{equation}
P = \frac{\tan^{-1}\left(\frac{209}{700 \times 3280.84 - 35000}\right) - \tan^{-1}\left(\frac{209}{700 \times 3280.84}\right)}{\tan^{-1}\left(\frac{209}{700 \times 3280.84}\right)} \times 100
\end{equation}

The calculated value for \( P \) is approximately \( 1.55\% \).

\section{Conclusion}

Therefore, for a satellite at \( 700 \) km above the Earth, the plane appears approximately \( 1.55\% \) larger to the satellite when the plane is at a height of \( 35000 \) feet versus sea level.


\end{document}