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%%% Template final.tex, version 1.0, November 2002 by Bruno Welfert %%%
%%% The file illustrates how to create tables and figures and       %%%
%%% illustrates the use of several style files                      %%%
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\documentclass[11pt]{report}
%=======================================================================
% Load style files
%=======================================================================
\usepackage{FYM}      % Style file for tests, finals
\usepackage{array}    % used to create matrices, tabular material
\usepackage{multicol} % defines multicols environment
\usepackage{graphicx}   % to insert .eps pictures
\usepackage{curves}   % to draw arcs of circles in figure environment
\usepackage{epic}     % to draw dotted lines, grids, ... in figure


%=======================================================================
% The following command (re)defines the spacing between the top of the
% page and the beginning of the text (1in+\topmargin). Setting \topmargin
% to a negative value moves the text up.
%=======================================================================
\renewcommand{\topmargin}{0in} % THIS MAY NEED TO BE ADJUSTED <=====
%=======================================================================


%=======================================================================
% adjust for your own test...
%=======================================================================
\renewcommand{\instructor}{Bruno Welfert}      % Instructor's name

\renewcommand{\course}{MAT 170}                % Course number

\renewcommand{\examnumber}{Exam III}           % Test type and number

\renewcommand{\examdate}{November 14, 2001}    % Date of test

\renewcommand{\examsemester}{Fall}             % Semester

\renewcommand{\examyear}{2002}                 % year

\renewcommand{\directions}{                    % Directions
    \begin{enumerate}
    \item Read each question carefully.
    \item Provide complete and well-organized answers.
          Circle your final answers.
    \item Always indicate your methods, and any way in which you used
          your calculator to solve a problem (this may include a hand-drawn
          sketch of a graph from the calculator screen).
    \item Answers without supporting work will be given zero credit.
          Partial credit is granted only if all work is shown.
          SHOW YOUR WORK!
    \item When sketching graphs, be sure to clearly denote your axes scales.
    \end{enumerate}
    }

\renewcommand{\exampolicy}{              % policy for calculators,...
No calculator that does symbolic algebra (e.g. TI-89, TI 92) is
allowed. Proctors reserve the right to randomly check calculators for
illegal programs and other methods of hiding information.}

\renewcommand{\covers}{Covers $\S$4.1, $\S$4.2, $\S$7.1} % Topics covered

% Optional information to appear at bottom of first page
% This should not exceed a few line, otherwise LaTeX may move it to
% second page. Here we include a few trigonometric formulas
\renewcommand{\optional}{
\textit{TRIGONOMETRIC FORMULAS:}\\[5pt]
\fbox{$\begin{array}{l}
\sin(x_1\pm x_2)=\sin x_1 \cos x_2 \pm \cos x_1 \sin x_2\\
\cos(x_1\pm x_2)=\cos x_1 \cos x_2 \mp \sin x_1 \sin x_2
\end{array}$}\hspace*{20pt}
\fbox{$\begin{array}{l}
\ds\left(\sin\frac{x}{2}\right)^2=\frac{1-\cos x}{2}\\
\ds\left(\cos\frac{x}{2}\right)^2=\frac{1+\cos x}{2}
\end{array}$}
}
%=======================================================================

\def\ds{\displaystyle}

\begin{document}

\draft       % Comment out for final version

\exampage    % Sets up the cover page of the test

\begin{enumerate}

\item
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% An example of tabular environment with constant cell width using array.sty
% the \PBS command is used to avoid problems with \\ at the end of each row
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\PBS}[1]{\let\temp=\\#1\let\\=\temp}%
\setlength{\extrarowheight}{3pt}%
Fill in the table below. All radians must be in terms of $\pi$, and
the values of $\sin\theta$ and $\cos\theta$ must be in exact form. In
other words, NO decimals! \textbf{[12 pts]}
\begin{center}
\begin{tabular}{
|>{\centering\bf}m{1in}<{}|>{\PBS\centering\bf}m{1in}|>{\PBS\centering\bf}m{1in}%
|>{\PBS\centering\bf}m{1in}|>{\PBS\centering\bf}m{1in}|}\hline%
Angle $\theta$\newline (degrees)& Angle $\theta$\newline (radians) &
Quadrant\newline of $\theta$ & $\sin\theta$ & $\cos\theta$ \\\hline% title row
$150^o$&&&&\\[20pt]\hline% row 1
&$\ds -\frac{3\pi}{4}$&&&\\[20pt]\hline% row 2
&$\ds \frac{5\pi}{3}$&&&\\[20pt]\hline% row 3
\end{tabular}
\end{center}

\item
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% An example of LaTeX picture environment for drawing simple figures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Sketch \textbf{one period} of the graph of the following functions on
the provided axes. Label the points corresponding to the beginning
and end of the period, the $x$-intercepts and the maximum and
minimum. Give the points in terms of $\pi$, no decimals. \textbf{[5
pts each]}
\begin{enumerate}
\item $\ds y = -3\cos\frac{x}{2}$\\\hspace*{2in}
\begin{picture}(200,150) % number are points
\put(0,75){\line(1,0){200}}\put(100,0){\line(0,1){150}}
\end{picture}
\item $\ds y = \sin(2x-\frac{\pi}{2})$\\\hspace*{2in}
\begin{picture}(200,150)
\put(0,75){\line(1,0){200}}\put(100,0){\line(0,1){150}}
\end{picture}
\end{enumerate}

\newpage

\item
Find all values of $t$ in the interval $[0,2\pi]$ that satisfy the
given equation. Give exact answers, no decimals. \textbf{[9 pts]}
$$ 2(\sin t)^2+\sin t - 1 = 0 $$
\vspace*{2in}

\item
Use the sum/difference formulas to verify the identity
$\sin(\pi-t)=\sin t$. \textbf{[5 pts]} \vspace*{1.5in}

\item
Find the exact value of the quantity (NO DECIMALS), or explain why it
is undefined. \textbf{[8 pts]}
\begin{enumerate}
\item $\arcsin(1/2)$\\[.5in]
\item $\arccos(-\sqrt{2}/2)$\\[.5in]
\item $\arcsin(\sin(3\pi/4))$\\[.5in]
\item $\tan(\arccos(2/5))$
\end{enumerate}

\newpage

\item
Algebraically verify the identity $\csc x - \sin x = \cot x \cos x$.
\textbf{[8 pts]} \vspace*{3in}

\item
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% An example of insertion of a .eps picture
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{[10 pts]} Find
\begin{enumerate}
\item a cosine function, and
\item a sine function
\end{enumerate}
whose graph matches the given curve. State period and amplitude.

\includegraphics[height=2.5in]{sine.eps}%
\hspace*{20pt}
\parbox[b]{3.5in}{
Period: \spot{1.5in}\\[.5in]
Amplitude: \spot{1.2in}\\[.5in]}\\[30pt]
cosine function: \spot{3in} \\[30pt]
sine function: \spot{3in}

\newpage

\item
Find \textbf{exact} values (NO DECIMALS) for $\cos\theta$,
$\tan\theta$, $\cot\theta$, $\sec\theta$, $\csc\theta$, $\cos
2\theta$, $\sin 2\theta$ is $\ds\sin\theta=-\frac{2}{3}$ and $\theta$
is in the fourth quandrant. (Do not solve for the angle $\theta$)
\textbf{[14 pts]}

\hfill $\cos\theta = $ \spot{2in} \\

\hfill $\tan\theta = $ \spot{2in} \\

\hfill $\cot\theta = $ \spot{2in} \\

\hfill $\sec\theta = $ \spot{2in} \\

\hfill $\csc\theta = $ \spot{2in} \\

\hfill $\cos 2\theta = $ \spot{2in} \\

\hfill $\sin 2\theta = $ \spot{2in}\\

\item
Consider the equation $\ds (\cos x)^2 = \frac{1}{9}$ on $[0,\pi]$.
\begin{enumerate}
\item
\textbf{[6 pts]} Solve the equation on the given interval, expressing
the solution for $x$ in terms of the inverse trigonometric
function.\\[1in]
\item
\textbf{[2 pts]} Use a calculator to give the approximate solution
from part (a) to 3 decimal places.\\[1in]
\item
\textbf{[4 pts]} Confirm your solution graphically. Explain by giving
the appropriate sketch.
\end{enumerate}

\newpage

\item
\textbf{[12 pts]} A boat leaves port and travels at a bearing $62^o$
south of east at 20 miles per hour. At the same time, a coast guard
cutter, located 26 miles due east of the port, heads towards the boat
at a constant bearing. The cutter intercepts the boat after one hour
(see picture below).
\begin{enumerate}
\item
What is the distance traveled by the cutter? \\
\setlength{\unitlength}{1in} % numbers are now inches
\begin{picture}(4,2.3)
%\put(0,0){\grid(4,2.5)(.5,.5)}%
\thicklines
\put(0,1.5){\includegraphics[height=.5in]{boat.eps}}
\put(1.5,1.6){\vector(-1,0){1.1}} \put(1.5,1.6){\vector(1,0){1.1}}
\put(2.7,1.5){\includegraphics[height=.5in]{boat.eps}}
\put(.4,1.5){\vector(1,-2){.25}} \put(2.6,1.5){\vector(-3,-2){.5}}
\put(.3,1.6){\arc(.25,-.4){58}} \put(.9,1.3){\makebox(0,0){$62^o$}}
\put(1.5,1.65){\makebox(0,0)[b]{$26$ miles}}
\dottedline{.05}(.4,1.5)(1,.3) \dottedline{.05}(2.6,1.5)(.8,.3)
\end{picture}
\vspace*{2.3in}
\item
What bearing did the cutter travel in order to intercept the boat?
\end{enumerate}

\end{enumerate}

\end{document}