\documentclass[11pt]{article}
\usepackage{precalc}
\usepackage{graphicx}
\usepackage{url}
\begin{document}
\assigntitle{13}{Triangle Laws}
\topic{the non-right triangles were getting sad}
The final installment of 2008 is also the final installment in our
study of trigonometry (but never fear, there will still be plenty of
triginometry involved in future assignments)! You already know how
trig functions relate to \emph{right} triangles; this week, we will
study two fundamental \term{triangle laws} that relate sine and cosine
to \emph{all} triangles. After all, this is \emph{trigonometry},
(``triangle measurement''), not \emph{orthotrigonometry} (``right
triangle measurement'')!
\section{The Law of Sines}
\label{sec:law-of-sines}
\begin{figure}[htp]
\centering
\includegraphics[scale=0.8]{diagrams/general-triangle.eps}
\caption{A triangle.}
\label{fig:general-triangle}
\end{figure}
\pref{fig:general-triangle} shows a generic triangle $ABC$. Notice how
it is labeled: the side opposite angle $A$ is labeled $a$; the side
opposite angle $B$ is labeled $b$, and so on. We will use the capital
letters to refer both to the vertices of the triangle and to the
angles at those vertices.
\topic{the Law of Sines}
The law of sines says this:
\begin{equation}
\label{eq:law-of-sines}
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\end{equation}
In words, for any triangle $ABC$, the ratios between the sines of the
angles and the corresponding opposite side lengths are all the same.
Note that this ratio may be different in different triangles. The Law
of Sines just says that in any particular triangle, all three ratios
will be the same.\footnote{In fact, the ratios are equal to the
diameter of the triangle's circumscribed circle! You probably don't
need to know that, but it is nifty.}
\begin{problem}
Check that the Law of Sines holds for the right triangle shown in
\pref{fig:rt-3-4-5}. Show your work.
\begin{figure}[htp]
\centering
\includegraphics{diagrams/rt-3-4-5.eps}
\caption{A 3-4-5 right triangle.}
\label{fig:rt-3-4-5}
\end{figure}
\end{problem}
\begin{problem}
Consider the triangle shown in \pref{fig:los-prob-1}.
\begin{subproblems}
\item What is $\theta$?
\item What is the length of side $q$? Show your work.
\end{subproblems}
\begin{figure}[htp]
\centering
\includegraphics{diagrams/los-prob-1.eps}
\caption{A triangle with given angle, side, angle (ASA)}
\label{fig:los-prob-1}
\end{figure}
\end{problem}
You can also read about something interesting I discovered about the Law of Sines
while writing this assignment here: \url{http://www.mathlesstraveled.com/?p=194}.
\section{The Law of Cosines}
\label{sec:law-of-cosines}
\topic{the Law of Cosines}
Again referring to \pref{fig:general-triangle}, the Law of Cosines
says that
\begin{equation}
\label{eq:law-of-cosines}
c^2 = a^2 + b^2 - 2ab \cos C.
\end{equation}
\begin{problem}
Explain the Law of Cosines in words.
\end{problem}
\begin{problem}
What happens to the Law of Cosines when $C$ is a right angle?
\end{problem}
\begin{problem}
Find the length of side $h$ of the triangle shown in
\pref{fig:loc-prob-1}. Show your work. Round your answer to three
decimal places.
\begin{figure}[htp]
\centering
\includegraphics{diagrams/loc-prob-1.eps}
\caption{A triangle with given side, angle, side (SAS)}
\label{fig:loc-prob-1}
\end{figure}
\end{problem}
\begin{problem}
In the triangle shown in \pref{fig:loc-prob-2}, find angle
$\theta$. Show your work. Round your answer to three decimal
places.
\begin{figure}[htp]
\centering
\includegraphics{diagrams/loc-prob-2.eps}
\caption{A triangle with three given sides (SSS)}
\label{fig:loc-prob-2}
\end{figure}
\end{problem}
\begin{problem}
In the triangle shown in \pref{fig:loc-prob-3}, find angle
$\theta$. Show your work. What does it mean that you get two
answers? Is this reasonable?
\begin{figure}[htp]
\centering
\includegraphics{diagrams/loc-prob-3.eps}
\caption{A triangle with given angle, side, side (\dots)}
\label{fig:loc-prob-3}
\end{figure}
\end{problem}
\end{document}