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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Tables of Trigonometric Functions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Feb 12 16:10:08 1997}
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\input{tcilatex}
\begin{document}


\section{Tables of Trigonometric Functions}

The usefulness of the trigonometric functions for solving practical problems
led to the compilation of tables of their values. The compilation of such
tables was a major challenge in past centuries. Since no table can list all
possible values of the trigonometric functions, each table listed the values
of the functions at certain selected points. Furthermore, since most of the
values of the functions must be expressed by nonterminating decimal
expansions people had to be content with approximations to the true values.
This was not a real hardship in practice---the numbers to which they applied
the functions were usually the result of a measurement which was itself an
approximation.

With computers and scientific calculators in our hands, these tables are no
longer useful. You still have to be content with approximations to the true
values of trigonometric functions, but you can almost instantaneously find
approximations to any degree you need.

Below are some very basic facts about trigonometric functions in tabular
form.

\subsubsection{Function Values for Basic Angles\label%
{function values for basic angles}}

\begin{center}
\begin{tabular}{c||cccccccc}
Degrees\smallskip & $0\unit{\UNICODE{0xb0}}$ & $30\unit{\UNICODE{0xb0}}$ & $%
45\unit{\UNICODE{0xb0}}$ & $60\unit{\UNICODE{0xb0}}$ & $90\unit{%
\UNICODE{0xb0}}$ & $180\unit{\UNICODE{0xb0}}$ & $270\unit{\UNICODE{0xb0}}$ & 
$360\unit{\UNICODE{0xb0}}$ \\ \hline
Radians & $0$ & $\dfrac{\pi }{6}\smallskip $ & $\dfrac{\pi }{4}$ & $\dfrac{%
\pi }{3}$ & $\dfrac{\pi }{2}$ & $\pi $ & $\dfrac{3\pi }{2}$ & $2\pi $ \\ 
\hline\hline
$\sin $ & $0$ & $\dfrac{1}{2}\smallskip $ & $\dfrac{\stackrel{}{\sqrt{2}}}{2}
$ & $\dfrac{\sqrt{3}}{2}$ & $1$ & $0$ & $-1$ & $0$ \\ 
$\cos $ & $1$ & $\dfrac{\sqrt{3}}{2}\smallskip $ & $\dfrac{\sqrt{2}}{2}$ & $%
\dfrac{1}{2}$ & $0$ & $-1$ & $0$ & $1$ \\ 
$\tan $ & $0$ & $\dfrac{\sqrt{3}}{3}$ & $1$ & $\sqrt{3}$ & $\infty $ & $0$ & 
$\infty $ & $0$ \\ 
$\cot $ & $\infty $ & $\sqrt{3}$ & $1$ & $\dfrac{\sqrt{3}}{3}$ & $0$ & $%
\infty $ & $0$ & $\infty $ \\ 
$\sec $ & $1$ & $\dfrac{2\sqrt{3}}{3}$ & $\sqrt{2}$ & $2$ & $\infty $ & $-1$
& $\infty $ & $1$ \\ 
$\csc $ & $\infty $ & $2$ & $\sqrt{2}$ & $\dfrac{2\sqrt{3}}{3}$ & $1$ & $%
\infty $ & $-1$ & $\infty $%
\end{tabular}
\bigskip
\end{center}

\subsubsection{Functions of Angles in Terms of Angles in First Quadrant\label%
{functions of angles in terms of angles in first quadrant}}

In this table, $n$ denotes an integer.

\begin{center}
\begin{tabular}{c||c|c|c|c|c|}
& $-\ \theta $ & $90\unit{\UNICODE{0xb0}}\pm \theta $ & $180\unit{%
\UNICODE{0xb0}}\pm \theta $ & $270\unit{\UNICODE{0xb0}}\pm \theta $ & $%
n\left( 360\right) \unit{\UNICODE{0xb0}}\pm \theta $ \\ \hline\hline
$\sin $ & $-\sin \theta $ & $+\cos \theta $ & $\mp \sin \theta $ & $-\cos
\theta $ & $\pm \sin \theta $ \\ 
$\cos $ & $+\cos \theta $ & $\mp \sin \theta $ & $-\cos \theta $ & $\pm \sin
\theta $ & $+\cos \theta $ \\ 
$\tan $ & $-\tan \theta $ & $\mp \cot \theta $ & $\pm \tan \theta $ & $\mp
\cot \theta $ & $\pm \tan \theta $ \\ 
$\cot $ & $-\cot \theta $ & $\mp \tan \theta $ & $\pm \cot \theta $ & $\mp
\tan \theta $ & $\pm \cot \theta $ \\ 
$\sec $ & $+\sec \theta $ & $\mp \csc \theta $ & $-\sec \theta $ & $\pm \csc
\theta $ & $+\sec \theta $ \\ 
$\csc $ & $-\csc \theta $ & $+\sec \theta $ & $\mp \csc \theta $ & $-\sec
\theta $ & $\pm \csc \theta $%
\end{tabular}
\end{center}

\strut

\subsubsection{Signs of the Trigonometric Functions\label%
{signs of the trigonometric functions}}

\begin{center}
\begin{tabular}{c|llllll}
\textsl{Quadrant} & $\sin $ & $\cos $ & $\tan $ & $\cot $ & $\sec $ & $\csc $
\\ \hline\hline
\textbf{I} & + & + & + & + & + & + \\ 
\textbf{II} & + & -- & -- & -- & -- & + \\ 
\textbf{III} & -- & -- & + & + & -- & -- \\ 
\textbf{IV} & -- & + & -- & -- & + & --
\end{tabular}
\bigskip
\end{center}

\subsubsection{Ranges of Trigonometric Functions\label%
{ranges of trigonometric functions}}

\begin{center}
\begin{tabular}{c|cccccc}
\textsl{Quadrant} & $\sin $ & $\cos $ & $\tan $ & $\cot $ & $\sec $ & $\csc $
\\ \hline\hline
\textbf{I} & $0\longrightarrow +1$ & \multicolumn{1}{|c}{$+1\longrightarrow
0 $} & \multicolumn{1}{|c}{$0\longrightarrow +\infty $} & 
\multicolumn{1}{|c}{$+\infty \longrightarrow 0$} & \multicolumn{1}{|c}{$%
+1\longrightarrow +\infty $} & \multicolumn{1}{|c}{$+\infty \longrightarrow
+1$} \\ 
\textbf{II} & $+1\longrightarrow 0$ & \multicolumn{1}{|c}{$0\longrightarrow
-1$} & \multicolumn{1}{|c}{$-\infty \longrightarrow 0$} & 
\multicolumn{1}{|c}{$0\longrightarrow -\infty $} & \multicolumn{1}{|c}{$%
-\infty \longrightarrow -1$} & \multicolumn{1}{|c}{$+1\longrightarrow
+\infty $} \\ 
\textbf{III} & $0\longrightarrow -1$ & \multicolumn{1}{|c}{$%
-1\longrightarrow 0$} & \multicolumn{1}{|c}{$0\longrightarrow +\infty $} & 
\multicolumn{1}{|c}{$+\infty \longrightarrow 0$} & \multicolumn{1}{|c}{$%
-1\longrightarrow -\infty $} & \multicolumn{1}{|c}{$-\infty \longrightarrow
-1$} \\ 
\textbf{IV} & $-1\longrightarrow 0$ & \multicolumn{1}{|c}{$0\longrightarrow
+1$} & \multicolumn{1}{|c}{$-\infty \longrightarrow 0$} & 
\multicolumn{1}{|c}{$0\longrightarrow -\infty $} & \multicolumn{1}{|c}{$%
+\infty \longrightarrow +1$} & \multicolumn{1}{|c}{$-1\longrightarrow
-\infty $}
\end{tabular}
\bigskip
\end{center}

\subsection{\ Related Reference Topics}

{\small \hyperref{Algebra}{}{}{Al_contents.tex}}

{\small \hyperref{Geometry}{}{}{GE_contents.tex}}

{\small \hyperref{Calculus}{}{}{CAL_contents.tex}}

{\small \hyperref{Tables of Trigonometric Facts}{}{}{TAB_trigonometry.tex}}%
\hfill

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}

{\small \hyperref{Function Definitions}{}{}{../../Help/DM5.tex}}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

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