\documentstyle[12pt]{report}

\setlength{\oddsidemargin}{0.0in}
\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{0.0in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\textheight}{9in}

\begin{document}

\noindent
{\large\bf Notes on Trigonometric Functions}

\bigskip

\noindent
{\large\bf Angle:}
\begin{itemize}
\item degree measure
\item radian measure
\item converting back and forth --- use proportion.
\end{itemize}

\noindent
{\bf Examples.}
Convert $30^{\circ}$ to radians. 
Convert $\pi/4$ radians to degrees.
How big is one radian?
\vspace{1.5in}

\noindent
{\large\bf Trigonometric Functions:}

\bigskip

\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(45,40)
\put(0,5){\circle*{1}}
\put(40,5){\circle*{1}}
\put(40,35){\circle*{1}}
\put(0,5){\line(1,0){40}}
\put(0,5){\line(4,3){40}}
\put(40,5){\line(0,1){30}}
\put(10,7){$\theta$}
\put(45,20){\rm opposite}
\put(15,0){\rm adjacent}
\put(0,25){\rm hypotenuse}
\end{picture}
\end{center}

\bigskip

\noindent
{\large\bf Important Definitions:} (Worth memorizing.)

\[\large
\begin{array}{rcl}
\sin(\theta) =& \frac{\rm opposite}{\rm hypotenuse} &\\
\cos(\theta) =& \frac{\rm adjacent}{\rm hypotenuse} &\\
\tan(\theta) =& \frac{\rm opposite}{\rm adjacent} &=
\frac{\sin(\theta)}{\cos(\theta)}\\
&&\\
\csc(\theta) =& \underline{\hspace{2in}} &=\frac{1}{\sin(\theta)} \\
\sec(\theta) =& \underline{\hspace{2in}} &=\frac{1}{\cos(\theta)} \\
\cot(\theta) =& \underline{\hspace{2in}} &=\frac{1}{\tan(\theta)} 
\end{array}\]

\bigskip

\noindent
Note that these definitions do not depend on the size of the
triangle, but only on the angle.

\break

\noindent
{\bf Exercise.}
Find all six trigonometric ratios for the following
angle.

\medskip


\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(45,40)
\put(0,5){\circle*{1}}
\put(40,5){\circle*{1}}
\put(40,35){\circle*{1}}
\put(0,5){\line(1,0){40}}
\put(0,5){\line(4,3){40}}
\put(40,5){\line(0,1){30}}
\put(10,7){$\theta$}
\put(45,20){\rm 3}
\put(20,0){\rm 4}
\put(15,20){\rm 5}
\end{picture}
\end{center}

\vspace{1.5in}



\noindent
{\bf Exercise.}
Compute the trig values that everyone is supposed to know 
``by heart'':

\bigskip

\[ \large
\begin{array}{|c||c|c|c|c|c|}
\hline
\theta &0 & \pi/6 & \pi/4 & \pi/3 & \pi/2\\
\hline
\sin(\theta) & \mbox{\hspace{.5in}} & \mbox{\hspace{.5in}} & 
\mbox{\hspace{.5in}} & \mbox{\hspace{.5in}} & \mbox{\hspace{.5in}} \\
\hline
\cos(\theta) &  &  &  &  &  \\
\hline
\tan(\theta) &  &  &  &  &  \\
\hline
\end{array}
\]


\vspace{.5in}



\noindent
The trigonometric functions are sometimes called
the ``circular functions'' for the following
reason: (Draw picture.)

\vspace{2.5in}

\noindent
Using this and the previous table, let's sketch
the graphs of $y = \sin(x), y = \cos(x)$ and $y = \tan(x)$.


\break

\noindent
{\large\bf Main Features of the Graph of $y=\sin(x)$:}

\bigskip

\begin{itemize}
\item Domain = \underline{\hspace{2in}}, 
Range = \underline{\hspace{2in}}.
\item Periodic of period \underline{\hspace{.75in}}.
In other words, $\sin(x) = \underline{\hspace{1.5in}}$.
\item Bounded above and below:  $-1\leq \sin(x) \leq 1$.
\item Odd.
\end{itemize}

\bigskip


\bigskip

\noindent
{\large\bf Main Features of the Graph of $y=\cos(x)$:}

\bigskip

\begin{itemize}
\item Domain = \underline{\hspace{2in}}, 
Range = \underline{\hspace{2in}}.
\item Periodic of period \underline{\hspace{.75in}}.
In other words, $\cos(x) = \underline{\hspace{1.5in}}$.
\item Bounded above and below:  $-1\leq \cos(x) \leq 1$.
\item Even.
\item The graph of $y = \cos(x)$ is a shifted copy of 
the graph of $y = \sin(x)$: in fact, $\cos(x) = \sin(x+\pi/2)$.
\item The graph of $y = \cos(x)$ is a reflection of
the graph of 
$y = \sin(x)$ through the line $x = \pi/2$: $\cos(x) =
\sin(\pi/2-x)$.
(Reflecting again shows that $\sin(x) = \cos(\pi/2 - x)$.)
\end{itemize}


\bigskip

\bigskip

\noindent
{\large\bf Main Features of the Graph of $y=\tan(x)$:}

\bigskip

\begin{itemize}
\item Domain = \underline{\hspace{2in}}, 
Range = \underline{\hspace{2in}}.
\item Periodic of period \underline{\hspace{.75in}}.
In other words, $\tan(x) = \underline{\hspace{1.5in}}$.
\item Unbounded near multiples of $\pi$.
\item Odd.
\end{itemize}

\bigskip



\end{document}
