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% Document Info
\author{lwaites}
\pdfinfo{
  /Title (solving-trigonometric-proofs.pdf)
  /Creator (Cheatography)
  /Author (lwaites)
  /Subject (Solving Trigonometric Proofs Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Solving Trigonometric Proofs Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{lwaites} via \textcolor{DarkBackground}{\uline{cheatography.com/125223/cs/28413/}}}
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\noindent
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}lwaites \\
  \uline{cheatography.com/lwaites} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 23rd June, 2021.\\
   Updated 23rd June, 2021.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\end{multicols}}




\begin{document}
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Solving Trigonometric Proofs}}  \tn
% Row 0
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\mymulticolumn{1}{x{8.4cm}}{{\bf{Method 1}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Pick one side of the equation (usually the most complicated side) and work with that side until it is equal to the other side} \tn 
% Row Count 4 (+ 3)
% Row 2
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\mymulticolumn{1}{x{8.4cm}}{{\bf{Method 2}}} \tn 
% Row Count 5 (+ 1)
% Row 3
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\mymulticolumn{1}{x{8.4cm}}{Work with both sides simultaneously until they are both equal to the same expression.} \tn 
% Row Count 7 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Helpful Techniques}}  \tn
% Row 0
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\mymulticolumn{1}{x{8.4cm}}{{\bf{Rewrite}}} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Rewrite the expression in terms of sine and cosine only} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\bf{Multiply by One}}} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Multiply the numerator and denominator of a rational expression by a carefully chosen "one"} \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{{\bf{Combine fractions}}} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Write sums and differences of rational expressions as a single fraction} \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\bf{Factor}}} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Factor trigonometric expressions, using "u-substitution" as needed} \tn 
% Row Count 12 (+ 3)
% Row 4
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\mymulticolumn{1}{x{8.4cm}}{{\bf{Pythagorean Theorem}}} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Look for combinations or portions of Pythagorean Identities.  Remember that you can multiply, divide, add or subtract the identity to get a new version.} \tn 
% Row Count 17 (+ 5)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\bf{Goal}}} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Always keep the goal in mind.  As you manipulate one side of the equation, keep the other side in mind.  Look for commonalities} \tn 
% Row Count 21 (+ 4)
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\par\addvspace{1.3em}

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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Verifying Trigonometric Identities}}  \tn
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\mymulticolumn{1}{p{8.4cm}}{ Video: http://youtu.be/Rf05H8ogHLg} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Pythagorean Identities}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{sin\textasciicircum{}2\textasciicircum{}x + cos\textasciicircum{}2\textasciicircum{}x = 1 \newline % Row Count 1 (+ 1)
~ sin\textasciicircum{}2\textasciicircum{}x = 1 - cos\textasciicircum{}2\textasciicircum{}x \newline % Row Count 2 (+ 1)
~      cos\textasciicircum{}2\textasciicircum{}x = 1 - sin\textasciicircum{}2\textasciicircum{}x \newline % Row Count 3 (+ 1)
1 + cot\textasciicircum{}2\textasciicircum{}x = csc\textasciicircum{}2\textasciicircum{}x \newline % Row Count 4 (+ 1)
~ cot\textasciicircum{}2\textasciicircum{}x = csc\textasciicircum{}2\textasciicircum{}x - 1 \newline % Row Count 5 (+ 1)
tan\textasciicircum{}2\textasciicircum{}x + 1 = sec\textasciicircum{}2\textasciicircum{}x \newline % Row Count 6 (+ 1)
~ tan\textasciicircum{}2\textasciicircum{}x = sec\textasciicircum{}2\textasciicircum{}x - 1% Row Count 7 (+ 1)
} \tn 
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\mymulticolumn{1}{x{8.4cm}}{Divide original Pythagorean identity by sin\textasciicircum{}2\textasciicircum{}x or cos\textasciicircum{}2\textasciicircum{}x to get other identities, subtract to get even more.}  \tn 
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\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Basic Trigonometric Functions}}  \tn
% Row 0
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sin (-x) = -sin (x) & tan (x) = sin (x) / cos (x) & csc (x) = 1/sin (x) \tn 
% Row Count 2 (+ 2)
% Row 1
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cos (-x) = cos (x) & cot (x) = cos (x) / sin (x) & sec (x) = 1/cos (x) \tn 
% Row Count 4 (+ 2)
% Row 2
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tan (-x) = -tan (x) &  & cot (x) = 1/tan (x) \tn 
% Row Count 6 (+ 2)
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Complementary Angle Identities}}  \tn
% Row 0
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\mymulticolumn{1}{x{8.4cm}}{sin (pi/2 - x) = cos (x)} \tn 
% Row Count 1 (+ 1)
% Row 1
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\mymulticolumn{1}{x{8.4cm}}{cos (pi/2 - x) = sin (x)} \tn 
% Row Count 2 (+ 1)
% Row 2
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\mymulticolumn{1}{x{8.4cm}}{tan (pi/2 - x) = cot (x)} \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.04 cm} x{4.96 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Sum and Difference Identities}}  \tn
% Row 0
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{\bf{sin (a +/- b)}}   = & sin (a) cos (b) +/- cos (a) sin (b) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
{\bf{cos (a +/- b)}}   = & cos (a) cos (b) -/+ sin (a) sin (b) \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{Use sin (a +/- b) / cos (a +/- b) to find {\bf{tan (a +/- b)}}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

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\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Double Angle Identities}}  \tn
% Row 0
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{\bf{sin (2x)}} & {\bf{cos (2x)}} &  \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
2 sin (x) cos (x) & cos\textasciicircum{}2\textasciicircum{} (x) - sin\textasciicircum{}2\textasciicircum{} (x) &  \tn 
% Row Count 3 (+ 2)
% Row 2
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 & 1 - 2 sin\textasciicircum{}2\textasciicircum{} (x) &  \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
 & 2 cos\textasciicircum{}2\textasciicircum{} (x) - 1 &  \tn 
% Row Count 5 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}