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\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
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% Document Info
\author{johanna e. . (sapphicpenguin)}
\pdfinfo{
  /Title (trigonometry.pdf)
  /Creator (Cheatography)
  /Author (johanna e. . (sapphicpenguin))
  /Subject (Trigonometry Cheat Sheet)
}

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\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
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% Header and Footer
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\fancyhead{} % Set header to blank
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Trigonometry Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{johanna e. . (sapphicpenguin)} via \textcolor{DarkBackground}{\uline{cheatography.com/68096/cs/17152/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}johanna e. . (sapphicpenguin) \\
  \uline{cheatography.com/sapphicpenguin} \\
        \uline{\seqsplit{www}.twitter.com/gay\_librarian}
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 21st September, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Quadratic Equation}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/sapphicpenguin_1537489779_kisspng-quadratic-equation-quadratic-function-quadratic-fo-formula-5ac24848a2fee6.2503047815226819286676.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.94103 cm} x{3.03597 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Discriminant}}  \tn
% Row 0
\SetRowColor{LightBackground}
b$^{\textrm{2}}$-4ac \textgreater{} 0 & 2 real roots \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
b$^{\textrm{2}}$-4ac = 0 & 1 repeated root \tn 
% Row Count 2 (+ 1)
% Row 2
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b$^{\textrm{2}}$-4ac \textless{} 0 & 2 imaginary roots \tn 
% Row Count 3 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.83689 cm} x{2.14011 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Supplementary \& Complementary Angles}}  \tn
% Row 0
\SetRowColor{LightBackground}
supplementary angles & add up to 180° \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
complementary angles & add up to 90° \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Pythagorean Theorem}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{a$^{\textrm{2}}$+b$^{\textrm{2}}$=c$^{\textrm{2}}$} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{45-45-90 triangles}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{a=b} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{c=a√2 or b√2} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{c/√2=a or b} \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{30-60-90 triangles}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{c=2b} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{a=b√3} \tn 
% Row Count 2 (+ 1)
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{when a triangle's 3 angles are 30°, 60°, and 90° \newline a = long leg \newline b = short leg \newline c = hypotenuse}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Area of a triangle}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{AΔ = (1/2)bh} \tn 
% Row Count 1 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.34379 cm} x{3.63321 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{SOH-CAH-TOA}}  \tn
% Row 0
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sine & opposite/hypotenuse \tn 
% Row Count 1 (+ 1)
% Row 1
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cosine & adjacent/hypotenuse \tn 
% Row Count 2 (+ 1)
% Row 2
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tangent & opposite/adjacent \tn 
% Row Count 3 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.59264 cm} x{3.38436 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{csc, sec, cot - the opposite of SOH-CAH-TOA}}  \tn
% Row 0
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cosecant & hypotenuse/opposite \tn 
% Row Count 1 (+ 1)
% Row 1
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secant & hypotenuse/adjacent \tn 
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% Row 2
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cotangent & adjacent/opposite \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.68758 cm} x{2.28942 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Circle Measurements}}  \tn
% Row 0
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area & πr$^{\textrm{2}}$ \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
diameter & 2r \tn 
% Row Count 2 (+ 1)
% Row 2
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circumference & 2πr OR πd \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
arc length & θr \tn 
% Row Count 4 (+ 1)
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{r = radius \newline d = diameter \newline θ = angle}  \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{3.18528 cm} x{1.79172 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Degrees \& Radians}}  \tn
% Row 0
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degrees to radians & θ·π/180 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
radians to degrees & θ·180/π \tn 
% Row Count 2 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Unit Circle}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/sapphicpenguin_1537490995_unit-circle-radians-degrees.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{2.4885 cm} p{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Unit Circle values}}  \tn
% Row 0
\SetRowColor{LightBackground}
sin & y \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
cos & x \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
tan & y/x \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
csc & 1/y \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
sec & 1/x \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
cot & x/y \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{for tan \& cot, only use the tops of the fractions}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.73735 cm} x{2.23965 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Coterminal Angles}}  \tn
% Row 0
\SetRowColor{LightBackground}
θ ± 360° & θ ± 2π \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.79632 cm} x{4.18068 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{y = a (sin) b (x - c) + d}}  \tn
% Row 0
\SetRowColor{LightBackground}
a & amplitude change \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
b & period change \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
c & horizontal change: + = left, - = right \tn 
% Row Count 4 (+ 2)
% Row 3
\SetRowColor{white}
d & vertical change: + = up, - + down \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\seqsplit{-sin(x)} & reflection across x-axis \tn 
% Row Count 7 (+ 2)
% Row 5
\SetRowColor{white}
\seqsplit{sin(-x)} & reflection across y=axis \tn 
% Row Count 9 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33919 cm} x{2.63781 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Amplitude \& Period}}  \tn
% Row 0
\SetRowColor{LightBackground}
amplitude & vertical \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
period & horizontal \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{both always positive \newline tan, cos, sec, cot: no amplitude}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.9862 cm} x{1.9908 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Inverses}}  \tn
% Row 0
\SetRowColor{LightBackground}
sin⁻$^{\textrm{1}}$(y) = x & sin(x) = y \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{csc(x) = 1/sin(x) \newline  \newline restricting range: usually I \& IV \newline except in cos⁻$^{\textrm{1}}$: I \& II}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Reciprocal Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
sinθ = 1/cscθ & cscθ = 1/sinθ \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
cosθ = 1/secθ & secθ = 1/cosθ \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
tanθ = 1/cotθ & cotθ = 1/tanθ \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Ratio Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
tanθ = sinθ/cosθ & cotθ = cosθ/sinθ \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Pythagorean Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{sin$^{\textrm{2}}$θ + cos$^{\textrm{2}}$θ = 1} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{tan$^{\textrm{2}}$θ + 1 = sec$^{\textrm{2}}$θ} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{1 + cos$^{\textrm{2}}$θ = csc$^{\textrm{2}}$θ} \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Double-Angle Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{sin2θ = 2sinθcosθ} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{tan2θ = 2tanθ / 1-tan$^{\textrm{2}}$θ} \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{cos2θ = cos$^{\textrm{2}}$θ - sin$^{\textrm{2}}$θ \newline cos2θ = 1 - 2sin$^{\textrm{2}}$θ \newline cos2θ = 2cos$^{\textrm{2}}$θ - 1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{some more identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{sin(a+b) = sin(a)cos(b) + cos(a)sin(b)} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{sin(a-b) = sin(a)cos(b) - cos(a)sin(b)} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{cos(a+b) = cos(a)cos(b) - sin(a)sin(b)} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{cos(a-b) = cos(a)cos(b) + sin(a)sin(b)} \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{tan(a+b) = tan(a)+tan(b) / 1-tan(a)tan(b) \newline tan(a-b) = tan(a)+tan(b) / 1+tan(a)tan(b)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}