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% Formulas for Trigonometry
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\title{Trigonometric Equivalences}
\author{}
\date{}
\begin{document}
\maketitle
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\begin{tabular}[t]{ccc}

\begin{minipage}{0.3\textwidth}
{Angles vs. Signs}
\begin{eqnarray*}
  \cos -\theta & = \cos\theta \\
  \sin -\theta & = -\sin\theta
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.3\textwidth}
{Periods}
\begin{eqnarray*}
  \cos \theta & = \sin(\theta + \frac{\pi}{2}) \\
  \cos n\pi & = \sin(n\pi + \frac{\pi}{2})
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.3\textwidth}
{Common Values}
\begin{eqnarray*}
\cos\frac{\pi}{6} & = \frac{\sqrt{3}}{2} \\
\sin\frac{\pi}{6} & = \frac{1}{2} \\
\tan\frac{\pi}{6} & = \frac{1}{\sqrt{3}}
\end{eqnarray*}
\end{minipage}

\end{tabular}




\interspace




%\section{Basic Relationships}

\begin{tabular}[t]{ccc}

\begin{minipage}{0.3\textwidth}
{Reciprocal Relationships}
\begin{eqnarray*}
  \csc\theta & = \frac{1}{\sin\theta} \\
  \sec\theta & = \frac{1}{\cos\theta} \\
  \cot\theta & = \frac{1}{\tan\theta}
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.3\textwidth}
{Quotient Relationships}
\begin{eqnarray*}
  \tan\theta & = \frac{\sin\theta}{\cos\theta} \\
  \cot\theta & = \frac{\cos\theta}{\sin\theta}
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.3\textwidth}
{Pythagorean Relationships}
\begin{eqnarray*}
  \sin^2\theta + \cos^2\theta = 1 \\
  1 + \tan^2\theta = \sec^2\theta \\
  1 + \cot^2\theta = \csc^2\theta
\end{eqnarray*}
\end{minipage}

\end{tabular}



\interspace



%\section{Addition/Subtraction Formulas}
\begin{tabular}[t]{cc}

\begin{minipage}{0.5\textwidth}
{Addition Formulas}
\begin{eqnarray*}
  \sin(\alpha + \beta) & = \sin\alpha \cos\beta + \cos\alpha \sin\beta \\
  \cos(\alpha + \beta) & = \cos\alpha \cos\beta - \sin\alpha \sin\beta \\
  \tan(\alpha + \beta) & = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta}
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.5\textwidth}
{Subtraction Formulas}
\begin{eqnarray*}
  \sin(\alpha - \beta) & = \sin\alpha \cos\beta - \cos\alpha \sin\beta \\
  \cos(\alpha - \beta) & = \cos\alpha \cos\beta + \sin\alpha \sin\beta \\
  \tan(\alpha - \beta) & = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \tan\beta}
\end{eqnarray*}
\end{minipage}
\end{tabular}



\interspace



\begin{tabular}[t]{cc}

\begin{minipage}{0.5\textwidth}
{Double-Angle Formulas}
\begin{eqnarray*}
  \sin 2\alpha & = 2 \sin\alpha \cos\alpha \\
  \cos 2\alpha & = \cos^2\alpha - \sin^2\alpha = \\
               & = 1 - 2\sin^2\alpha = 2\cos^2\alpha -1 \\
  \tan 2\alpha & = \frac{2 \tan\alpha}{1 - \tan^2\alpha}
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.5\textwidth}
{Half-Angle Formulas}
\begin{eqnarray*}
  \sin \frac{1}{2}\theta & = \pm \sqrt{\frac{1 - \cos\theta}{2}} \\
  \cos \frac{1}{2}\theta & = \pm \sqrt{\frac{1 + \cos\theta}{2}} \\
  \tan \frac{1}{2}\theta & = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} =
    \frac{\sin\theta}{1 + \cos\theta} =
    \frac{1 - \cos\theta}{\sin\theta}
\end{eqnarray*}
\end{minipage}

\end{tabular}




\interspace




\begin{tabular}[t]{cc}

\begin{minipage}{0.5\textwidth}
{Products of Sines and Cosines}
\begin{eqnarray*}
  \sin\alpha \cos\beta & = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)] \\
  \cos\alpha \sin\beta & = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)] \\
  \cos\alpha \cos\beta & = \frac{1}{2}[\cos(\alpha + \beta) + \cos(\alpha - \beta)] \\
  \sin\alpha \sin\beta & = \frac{1}{2}[\cos(\alpha + \beta) - \cos(\alpha - \beta)]
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.5\textwidth}
{Sum and Difference of Sines and Cosines}
\begin{eqnarray*}
  \sin\alpha + \sin\beta & = 2 \sin\frac{1}{2}(\alpha + \beta) \cos\frac{1}{2}(\alpha - \beta) \\
  \sin\alpha - \sin\beta & = 2 \cos\frac{1}{2}(\alpha + \beta) \sin\frac{1}{2}(\alpha - \beta) \\
  \cos\alpha + \cos\beta & = 2 \cos\frac{1}{2}(\alpha + \beta) \cos\frac{1}{2}(\alpha - \beta) \\
  \sin\alpha + \sin\beta & = 2 \sin\frac{1}{2}(\alpha + \beta) \sin\frac{1}{2}(\alpha - \beta)
\end{eqnarray*}
\end{minipage}
\end{tabular}



\interspace



\begin{tabular}[t]{cc}
\begin{minipage}{0.5\textwidth}
{Law of Sines}
\begin{displaymath}
  \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\end{displaymath}
\begin{displaymath}
  \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
\end{displaymath}
\end{minipage}

\begin{minipage}{0.5\textwidth}
{Law of Cosines}
\begin{eqnarray*}
  a^2 = b^2 + c^2 - 2bc\cos A \\
  b^2 = a^2 + c^2 - 2ac\cos B \\
  c^2 = a^2 + b^2 - 2ab\cos C
\end{eqnarray*}
\end{minipage}

\end{tabular}



{\vfill\hfill{\tiny Author: Martin Blais, 2009.
This work is licensed under the Creative Commons
``Attribution - Non-Commercial - Share-Alike'' license.}}
\end{document}