\documentclass[11pt]{book}

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\addtolength{\textwidth}{.6in}

\addtolength{\topmargin}{-.6in}
\addtolength{\textheight}{1.5in}

\usepackage{amsmath}

\usepackage{fancyhdr}

\usepackage{tkz-euclide}
\usetkzobj{all}

\newcommand{\Problem}[1]{\item[\fbox{\parbox{.25in}{\hfil{#1}\hfil}}]}


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\lhead{\textbf{MAC 1147 - Quiz 8 Key}}
\rhead{\textbf{Name:}\hspace{6cm}}
\cfoot{}
\pagestyle{fancy}
\setlength{\headsep}{.25in}

\begin{document}
\thispagestyle{fancy}


%% =================
%%          Instructions
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\textbf{For full credit, you must \underline{show all work} and \underline{circle} your final answer.}\\


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\begin{enumerate}


	\Problem{1} The point $(3, \sqrt7 )$ is on the terminal side of an angle in standard position. Determine the values of the six trigonometric functions of this angle. \\	

Graphing the point $3, \sqrt7)$ on a cartesian graph results in the following triangle:



	\begin{tikzpicture}[thick]
\coordinate (O) at (0,0);
\coordinate (A) at (3,0);
\coordinate (B) at (3,2.6);
\draw (O)--(A)--(B)--cycle;

\tkzLabelSegment[below=2pt](O,A){3}
\tkzLabelSegment[left=2pt](O,B){x}
\tkzLabelSegment[above right=2pt](A,B){$\sqrt7$}

\tkzLabelAngle[pos = 0.6](B,O,A){$\theta$}

\end{tikzpicture}\\
To find x we use the pythagorean theorem and find that $x = 4$, so we have the new picture: \\

\begin{tikzpicture}[thick]
\coordinate (O) at (0,0);
\coordinate (A) at (3,0);
\coordinate (B) at (3,2.6);
\draw (O)--(A)--(B)--cycle;

\tkzLabelSegment[below=2pt](O,A){3}
\tkzLabelSegment[left=2pt](O,B){4}
\tkzLabelSegment[above right=2pt](A,B){$\sqrt7$}


\tkzLabelAngle[pos = 0.6](B,O,A){$\theta$}

\end{tikzpicture}\\

Now we can define the trigonemetric functions: \\

$sin( \theta)  = \frac {\sqrt7}{4}    $ $   csc( \theta)  = \frac {4}{\sqrt7} $\\
$cos( \theta)  = \frac {3}{4}    $ $   sec( \theta)  = \frac {4}{3} $\\
$tan( \theta)  = \frac {\sqrt7}{3}   $ $  cot( \theta)  = \frac {3}{\sqrt7} $\\

	\vfill

	\Problem{2} Find the period and amplitude of the following function and sketch the graph. Label the x-intercepts: \\
	$f(x) = 3cos(2x)$\\

Amplitude $= |3| = 3 $ \\
Period $ = \frac{2\pi}{2} = 2 $\\
Graph excluded.

	\vfill

	\Problem{3} Find x and r:\\
\\
	\begin{tikzpicture}[thick]
\coordinate (O) at (0,0);
\coordinate (A) at (4,0);
\coordinate (B) at (0,2);
\draw (O)--(A)--(B)--cycle;

\tkzLabelSegment[below=2pt](O,A){$30 \sqrt3$}
\tkzLabelSegment[left=2pt](O,B){x}
\tkzLabelSegment[above right=2pt](A,B){r}


\tkzMarkAngle[fill= orange,size=0.8cm,%
opacity=.4](B,A,O)
\tkzLabelAngle[pos = 0.6](B,A,O){$30$}

\tkzMarkAngle[fill= orange,size=0.7cm,%
opacity=.4](O,B,A)
\tkzLabelAngle[pos = 0.5](O,B,A){}


\end{tikzpicture}

We can find the lengths of the sides by evaluating the following: \\
$cos(30) = \frac{30 \sqrt3}{r}$, so $ r = \frac{30 \sqrt3}{cos(30)} = \frac{30 \sqrt3}{\frac{\sqrt3}{2}} = 60 $ \\
$tan(30) = \frac{x}{30 \sqrt3}$, so $ x = 30 \sqrt3 *tan(30) = 30 \sqrt3 * \frac{1}{\sqrt3 } = 30 $

	\vfill 	

\end{enumerate}


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