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\title{Cheat Sheet 2: Sine Rule and Cosine Rule}
\author{Simon Harris}
\date{\today}
\maketitle


\section{The Sine Rule}

For any triangle, given the sides $a$, $b$ and $c$ and their corresponding opposite angles $A$, $B$ and $C$:

\begin{equation*}
\frac{sin\ A}{a} = \frac{sin\ B}{b} = \frac{sin\ C}{c}
\end{equation*}

\begin{equation*}
\frac{a}{sin\ A} = \frac{b}{sin\ B} = \frac{c}{sin\ C}
\end{equation*}

So, given two sides and a corresponding angle, or two angles and a corresponding side, the triangle can be solved.

\section{The Cosine Rule}

Given two sides plus the angle between them:\footnote{This is a generalisation of Pythagoras' Theorem, to which it reduces if the angle is 90\degree}

\begin{align*}
a^{2} =& \ b^{2} + c^{2} - 2bc\ cos\ A\\
b^{2} =& \ c^{2} + a^{2} - 2ca\ cos\ B\\
c^{2} =& \ a^{2} + b^{2} - 2ab\ cos\ C
\end{align*}

Given 3 sides but no angle, this form is more convenient:

\begin{equation*}
cos\ A = \frac{b^{2} + c^{2} - a^{2}}{2bc},\ cos\ B = \frac{c^{2} + a^{2} - b^{2}}{2ca},\ cos\ C = \frac{a^{2} + b^{2} - c^{2}}{2ab}
\end{equation*}


\section{The General Component Form}

For converting a vector from geometric to component form:

\begin{equation*}
\textbf{a} = \abs{\textbf{a}}\ cos\ \theta\textbf{i} + \abs{\textbf{a}}\ sin\ \theta\textbf{j}
\end{equation*}

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