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% Document Info
\author{CROSSANT (CROSSANT)}
\pdfinfo{
  /Title (trigonometric-properties-and-identities.pdf)
  /Creator (Cheatography)
  /Author (CROSSANT (CROSSANT))
  /Subject (Trigonometric Properties and Identities Cheat Sheet)
}

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\noindent
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\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{Trigonometric Properties and Identities Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{CROSSANT (CROSSANT)} via \textcolor{DarkBackground}{\uline{cheatography.com/186482/cs/39959/}}}
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\noindent
\begin{multicols}{3}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}CROSSANT (CROSSANT) \\
  \uline{cheatography.com/crossant} \\
  \end{tabulary}
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\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 26th August, 2024.\\
   Updated 27th December, 2024.\\
   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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\begin{document}
\raggedright
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\begin{multicols*}{4}

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\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Circular Functions Definitions}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Name}} & {\bf{Right-Triangle Definition}} & {\bf{Domain}} & {\bf{Range}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Sine Function & sin(θ)=o/h & (-∞,∞) & {[}-1,1{]} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Cosine Function & cos(θ)=a/h & (-∞,∞) & {[}-1,1{]} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Tangent Function & tan(θ)=o/a & \{θ|θ≠π/2±πn\} & (-∞,∞) \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Cosecant Function & csc(θ)=h/o & \{θ|θ≠±πn\} & (-∞,-1{]}∪{[}1,∞) \tn 
% Row Count 12 (+ 3)
% Row 5
\SetRowColor{white}
Secant Function & sec(θ)=h/a & \{θ|θ≠π/2±πn\} & (-∞,-1{]}∪{[}1,∞) \tn 
% Row Count 15 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Cotangent Function & cot(θ)=a/o & \{θ|θ≠±πn\} & (-∞,∞) \tn 
% Row Count 18 (+ 3)
% Row 7
\SetRowColor{white}
Inverse Sine Function & arcsin(o/h)=θ & {[}-1,1{]} & {[}-π/2,π/2{]} \tn 
% Row Count 21 (+ 3)
% Row 8
\SetRowColor{LightBackground}
Inverse Cosine Function & arccos(a/h)=θ & {[}-1,1{]} & {[}0,π{]} \tn 
% Row Count 24 (+ 3)
% Row 9
\SetRowColor{white}
Inverse Tangent Function & arctan(o/a)=θ & (-∞,∞) & (-1,1) \tn 
% Row Count 27 (+ 3)
% Row 10
\SetRowColor{LightBackground}
Inverse Cosecant Function & arccsc(h/o)=θ & (-∞,-1)∪(1,∞) & {[}-π/2,0)∪(0,π/2{]} \tn 
% Row Count 30 (+ 3)
\end{tabularx}
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\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Circular Functions Definitions (cont)}}  \tn
% Row 11
\SetRowColor{LightBackground}
Inverse Secant Function & arcsec(h/a)=θ & (-∞,-1)∪(1,∞) & {[}0,π/2)∪(π/2,π{]} \tn 
% Row Count 3 (+ 3)
% Row 12
\SetRowColor{white}
Inverse Cotangent Function & arccot(a/o)=θ & (-∞,∞) & (0,1) \tn 
% Row Count 6 (+ 3)
% Row 13
\SetRowColor{LightBackground}
Circular Euler Relation & e\textasciicircum{}±iθ\textasciicircum{}=cos(θ)±isin(θ) &  &  \tn 
% Row Count 9 (+ 3)
% Row 14
\SetRowColor{white}
De Moivre's Theorem & e\textasciicircum{}inθ\textasciicircum{}=(cos(θ)+isin(θ))\textasciicircum{}n\textasciicircum{}=cos(nθ)+isin(nθ) &  &  \tn 
% Row Count 13 (+ 4)
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\SetRowColor{LightBackground}
\mymulticolumn{4}{x{3.833cm}}{n ∈ ℕ\textasciitilde{}1\textasciitilde{} = \{1,2,3,4,5,...\} \newline "h" is the "hypotenuse" leg of a right triangle. It is directly across the right (90°) angle, and it has the longest length of the three sides. \newline "o" is the "opposite" leg of a right triangle. It is directly across the chosen angle θ. \newline "a" is the "adjacent" leg of a right triangle. It is the leg that is neither the hypotenuse leg, nor the opposite leg. \newline By the Pythagorean theorem, o\textasciicircum{}2\textasciicircum{}+a\textasciicircum{}2\textasciicircum{}=h\textasciicircum{}2\textasciicircum{}}  \tn 
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% Row 0
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{\bf{Name}} & {\bf{Exponential Definition}} & {\bf{Domain}} & {\bf{Range}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Hyperbolic Sine Function & sinh(θ)=(e\textasciicircum{}θ\textasciicircum{}-e\textasciicircum{}-θ\textasciicircum{})/2 & (-∞,∞) & (-∞,∞) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Hyperbolic Cosine Function & cosh(θ)=(e\textasciicircum{}θ\textasciicircum{}+e\textasciicircum{}-θ\textasciicircum{})/2 & (-∞,∞) & {[}1,∞) \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
Hyperbolic Tangent Function & tanh(θ)=(e\textasciicircum{}θ\textasciicircum{}-e\textasciicircum{}-θ\textasciicircum{})/(e\textasciicircum{}θ\textasciicircum{}+e\textasciicircum{}-θ\textasciicircum{}) & (-∞,∞) & (-1,1) \tn 
% Row Count 10 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Hyperbolic Cosecant Function & csch(θ)=2/(e\textasciicircum{}θ\textasciicircum{}-e\textasciicircum{}-θ\textasciicircum{}) & (-∞,0)∪(0,∞) & (-∞,0)∪(0,∞) \tn 
% Row Count 13 (+ 3)
% Row 5
\SetRowColor{white}
Hyperbolic Secant Function & sech(θ)=2/(e\textasciicircum{}θ\textasciicircum{}+e\textasciicircum{}-θ\textasciicircum{}) & (-∞,∞) & (0,1{]} \tn 
% Row Count 16 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Hyperbolic Cotangent Function & coth(θ)=(e\textasciicircum{}θ\textasciicircum{}+e\textasciicircum{}-θ\textasciicircum{})/(e\textasciicircum{}θ\textasciicircum{}-e\textasciicircum{}-θ\textasciicircum{}) & (-∞,0)∪(0,∞) & (-∞,-1)∪(1,∞) \tn 
% Row Count 20 (+ 4)
% Row 7
\SetRowColor{white}
Inverse Hyperbolic Sine Function & arcsinh(x)=ln(x+{\bf{√}}(x\textasciicircum{}2\textasciicircum{}+1)) & (-∞,∞) & (-∞,∞) \tn 
% Row Count 23 (+ 3)
% Row 8
\SetRowColor{LightBackground}
Inverse Hyperbolic Cosine Function & arccosh(x)=ln(x+{\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1)) & {[}1,∞) & {[}0,∞) \tn 
% Row Count 26 (+ 3)
% Row 9
\SetRowColor{white}
Inverse Hyperbolic Tangent Function & arctanh(x)=$\frac{1}{2}$ln((1+x)/(1-x)) & (-1,1) & (-∞,∞) \tn 
% Row Count 29 (+ 3)
% Row 10
\SetRowColor{LightBackground}
Inverse Hyperbolic Cosecant Function & arccsch(x)=ln((1±{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{}))/x) & (-∞,0)∪(0,∞) & (-∞,0)∪(0,∞) \tn 
% Row Count 32 (+ 3)
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Inverse Hyperbolic Secant Function & arcsech(x)=ln((1+{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}))/x) & (0,1{]} & {[}0,∞) \tn 
% Row Count 3 (+ 3)
% Row 12
\SetRowColor{white}
Inverse Hyperbolic Cotangent Function & arccoth(x)=$\frac{1}{2}$ln((x+1)/(x-1)) & (-∞,-1)∪(1,∞) & (-∞,0)∪(0,∞) \tn 
% Row Count 7 (+ 4)
% Row 13
\SetRowColor{LightBackground}
Hyperbolic Euler Relation & e\textasciicircum{}±θ\textasciicircum{}=cosh(θ)±sinh(θ) &  &  \tn 
% Row Count 10 (+ 3)
% Row 14
\SetRowColor{white}
De Moivre's Theorem \seqsplit{(Hyperbolic)} & e\textasciicircum{}nθ\textasciicircum{}=(cosh(θ)+sinh(θ))\textasciicircum{}n\textasciicircum{}=cosh(nθ)+sinh(nθ)) &  &  \tn 
% Row Count 14 (+ 4)
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\hhline{>{\arrayrulecolor{DarkBackground}}----}
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\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Complex Definitions}}  \tn
% Row 0
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{\bf{Name}} & {\bf{Complex Relation}} & {\bf{Circular-Hyperbolic Relation}} &  \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Complex Sine & sin(z)=(e\textasciicircum{}iz\textasciicircum{}-e\textasciicircum{}-iz\textasciicircum{})/2i & \seqsplit{sin(z)=-isinh(iz)} &  \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Complex Cosine & cos(z)=(e\textasciicircum{}iz\textasciicircum{}+e\textasciicircum{}-iz\textasciicircum{})/2 & \seqsplit{cos(z)=cosh(iz)} &  \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
Complex Tangent & tan(z)=-i(e\textasciicircum{}iz\textasciicircum{}-e\textasciicircum{}-iz\textasciicircum{})/(e\textasciicircum{}iz\textasciicircum{}+e\textasciicircum{}-iz\textasciicircum{}) & \seqsplit{tan(z)=-itanh(iz)} &  \tn 
% Row Count 13 (+ 4)
% Row 4
\SetRowColor{LightBackground}
Complex Cosecant & csc(z)=2i/(e\textasciicircum{}iz\textasciicircum{}-e\textasciicircum{}-iz\textasciicircum{}) & \seqsplit{csc(z)=icsch(iz)} &  \tn 
% Row Count 16 (+ 3)
% Row 5
\SetRowColor{white}
Complex Secant & sec(z)=2/(e\textasciicircum{}iz\textasciicircum{}+e\textasciicircum{}-iz\textasciicircum{}) & \seqsplit{sec(z)=sech(iz)} &  \tn 
% Row Count 19 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Complex Cotangent & cot(z)=i(e\textasciicircum{}iz\textasciicircum{}+e\textasciicircum{}-iz\textasciicircum{})/(e\textasciicircum{}iz\textasciicircum{}-e\textasciicircum{}-iz\textasciicircum{}) & \seqsplit{cot(z)=icoth(iz)} &  \tn 
% Row Count 23 (+ 4)
% Row 7
\SetRowColor{white}
Complex Inverse Sine & arcsin(z)=-iln(iz±{\bf{√}}(1-z\textasciicircum{}2\textasciicircum{})) & \seqsplit{arcsin(z)=-iarcsinh(iz)} &  \tn 
% Row Count 27 (+ 4)
% Row 8
\SetRowColor{LightBackground}
Complex Inverse Cosine & arccos(z)=-iln(z±i{\bf{√}}(1-z\textasciicircum{}2\textasciicircum{})) & \seqsplit{arccos(z)=±iarccosh(z)} &  \tn 
% Row Count 31 (+ 4)
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% Row 9
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Complex Inverse Tangent & \seqsplit{arctan(z)=(i/2)ln((i+z)/(i-z))} & \seqsplit{arctan(z)=-iarctanh(iz)} &  \tn 
% Row Count 3 (+ 3)
% Row 10
\SetRowColor{white}
Complex Inverse Cosecant & arccsc(z)=-iln((i+{\bf{√}}(z\textasciicircum{}2\textasciicircum{}-1))/z) & \seqsplit{arccsc(z)=iarccsch(iz)} &  \tn 
% Row Count 7 (+ 4)
% Row 11
\SetRowColor{LightBackground}
Complex Inverse Secant & arcsec(z)=-iln((1+{\bf{√}}(1-z\textasciicircum{}2\textasciicircum{}))/z) & \seqsplit{arcsec(z)=±iarcsech(z)} &  \tn 
% Row Count 11 (+ 4)
% Row 12
\SetRowColor{white}
Complex Inverse Cotangent & \seqsplit{arccot(z)=-(i/2)ln((z+i)/(z-i))} & \seqsplit{arccot(z)=±iarccoth(iz)} &  \tn 
% Row Count 15 (+ 4)
% Row 13
\SetRowColor{LightBackground}
Complex Hyperbolic Sine & None & \seqsplit{sinh(z)=-isin(iz)} &  \tn 
% Row Count 17 (+ 2)
% Row 14
\SetRowColor{white}
Complex Hyperbolic Cosine & None & \seqsplit{cosh(z)=cos(iz)} &  \tn 
% Row Count 19 (+ 2)
% Row 15
\SetRowColor{LightBackground}
Complex Hyperbolic Tangent & None & \seqsplit{tanh(z)=-itan(iz)} &  \tn 
% Row Count 21 (+ 2)
% Row 16
\SetRowColor{white}
Complex Hyperbolic Cosecant & None & \seqsplit{csch(z)=icsc(iz)} &  \tn 
% Row Count 24 (+ 3)
% Row 17
\SetRowColor{LightBackground}
Complex Hyperbolic Secant & None & \seqsplit{sech(z)=sec(iz)} &  \tn 
% Row Count 26 (+ 2)
% Row 18
\SetRowColor{white}
Complex Hyperbolic Cotangent & None & \seqsplit{coth(z)=icot(iz)} &  \tn 
% Row Count 29 (+ 3)
% Row 19
\SetRowColor{LightBackground}
Complex Inverse Hyperbolic Sine & None & \seqsplit{arcsinh(z)=-iarcsin(iz)} &  \tn 
% Row Count 32 (+ 3)
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% Row 20
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Complex Inverse Hyperbolic Cosine & None & \seqsplit{arccosh(z)=±iarccos(z)} &  \tn 
% Row Count 3 (+ 3)
% Row 21
\SetRowColor{white}
Complex Inverse Hyperbolic Tangent & None & \seqsplit{arctanh(z)=-iarctan(iz)} &  \tn 
% Row Count 6 (+ 3)
% Row 22
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Complex Inverse Hyperbolic Cosecant & None & \seqsplit{arccsch(z)=-iarccsc(iz)} &  \tn 
% Row Count 9 (+ 3)
% Row 23
\SetRowColor{white}
Complex Inverse Hyperbolic Secant & None & \seqsplit{arcsech(z)=±iarcsec(z)} &  \tn 
% Row Count 12 (+ 3)
% Row 24
\SetRowColor{LightBackground}
Complex Inverse Hyperbolic Cotangent & None & \seqsplit{arccoth(z)=-iarccot(iz)} &  \tn 
% Row Count 15 (+ 3)
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\mymulticolumn{4}{x{3.833cm}}{i={\bf{√}}(-1) \newline z is a complex variable of the form a+bi, where a and b are real numbers, and i is the imaginary unit}  \tn 
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% Row 1
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% Row Count 5 (+ 1)
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% Row Count 6 (+ 0)
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% Row Count 8 (+ 1)
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% Row Count 9 (+ 1)
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% Row Count 10 (+ 0)
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% Row Count 13 (+ 1)
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% Row Count 14 (+ 1)
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{} \tn 
% Row Count 14 (+ 0)
% Row 17
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{{\bf{5π/3}}} \tn 
% Row Count 15 (+ 1)
% Row 18
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{{\bf{7π/4}}} \tn 
% Row Count 16 (+ 1)
% Row 19
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{{\bf{11π/6}}} \tn 
% Row Count 17 (+ 1)
% Row 20
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{{\bf{2π}}} \tn 
% Row Count 18 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{The coordinates (cos(θ), sin(θ)) represent x and y coordinates of θ on the unit circle x\textasciicircum{}2\textasciicircum{}+y\textasciicircum{}2\textasciicircum{}=1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.55293 cm} p{0.71091 cm} p{0.68458 cm} p{0.68458 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Circular Compositional Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Composition}} & {\bf{sin(x)}} & {\bf{cos(x)}} & {\bf{tan(x)}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
{\bf{arcsin(x)}} & x & {\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) & x/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
{\bf{arccos(x)}} & {\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) & x & {\bf{√}}(1-x\textasciicircum{}2\textasciicircum{})/x \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
{\bf{arctan(x)}} & x/{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{}) & 1/{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{}) & x \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
{\bf{arccsc(x)}} & 1/x & {\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1)/|x| & ±1/{\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1) \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
{\bf{arcsec(x)}} & {\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1)/|x| & 1/x & ±{\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1) \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
{\bf{arccot(x)}} & 1/{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{}) & x/{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{}) & 1/x \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\SetRowColor{LightBackground}
\mymulticolumn{4}{x{3.833cm}}{Each composition is valid on different domains}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.57926 cm} p{0.65825 cm} p{0.73724 cm} p{0.65825 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{3.833cm}}{\bf\textcolor{white}{Hyperbolic Compositional Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Composition}} & {\bf{sinh(x)}} & {\bf{cosh(x)}} & {\bf{tanh(x)}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
{\bf{arcsinh(x)}} & x & {\bf{√}}(1+x\textasciicircum{}2\textasciicircum{}) & x/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
{\bf{arccosh(x)}} & {\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1) & x & {\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1)/x \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
{\bf{arctanh(x)}} & x/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) & 1/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) & x \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
{\bf{arccsch(x)}} & 1/x & {\bf{√}}(x\textasciicircum{}2\textasciicircum{}+1)/|x| & 1/{\bf{√}}(x\textasciicircum{}2\textasciicircum{}+1) \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
{\bf{arcsech(x)}} & {\bf{√}}(1-x\textasciicircum{}2\textasciicircum{})/x & 1/x & {\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
{\bf{arccoth(x)}} & x/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) & |x|/{\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1) & 1/x \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\SetRowColor{LightBackground}
\mymulticolumn{4}{x{3.833cm}}{Each composition is valid on different domains}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.61351 cm} x{1.81949 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular Quotient \& Reciprocal Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Tangent Quotient & \seqsplit{tan(θ)=sin(θ)/cos(θ)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cotangent Quotient & \seqsplit{cot(θ)=cos(θ)/sin(θ)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 4 (+ 0)
% Row 3
\SetRowColor{white}
Sine Reciprocal & sin(θ)=1/csc(θ) \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Cosine Reciprocal & cos(θ)=1/sec(θ) \tn 
% Row Count 6 (+ 1)
% Row 5
\SetRowColor{white}
Tangent Reciprocal & tan(θ)=1/cot(θ) \tn 
% Row Count 7 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Cosecant Reciprocal & csc(θ)=1/sin(θ) \tn 
% Row Count 9 (+ 2)
% Row 7
\SetRowColor{white}
Secant Reciprocal & sec(θ)=1/cos(θ) \tn 
% Row Count 10 (+ 1)
% Row 8
\SetRowColor{LightBackground}
Cotangent Reciprocal & cot(θ)=1/tan(θ) \tn 
% Row Count 12 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{All the following identities are true for values that do not cause division by zero}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.81949 cm} x{1.61351 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Cofunctional Phase Shift Properties}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Complimentary & \seqsplit{sin(θ)=cos(π/2-θ)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Sine Supplementary & \seqsplit{sin(θ)=sin(π-θ)} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Cosine Complimentary & \seqsplit{cos(θ)=sin(π/2-θ)} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
Cosine Supplementary & \seqsplit{cos(θ)=-cos(π-θ)} \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Tangent Complimentary & \seqsplit{tan(θ)=cot(π/2-θ)} \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
Tangent Supplementary & \seqsplit{tan(θ)=-tan(πn-θ)} \tn 
% Row Count 11 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Cosecant Complimentary & \seqsplit{csc(θ)=sec(π/2-θ)} \tn 
% Row Count 13 (+ 2)
% Row 7
\SetRowColor{white}
Cosecant Supplementary & \seqsplit{csc(θ)=csc(π-θ)} \tn 
% Row Count 15 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Secant Complimentary & \seqsplit{sec(θ)=csc(π/2-θ)} \tn 
% Row Count 17 (+ 2)
% Row 9
\SetRowColor{white}
Secant Supplementary & \seqsplit{sec(θ)=-sec(π-θ)} \tn 
% Row Count 19 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Cotangent Complimentary & \seqsplit{cot(θ)=tan(π/2-θ)} \tn 
% Row Count 21 (+ 2)
% Row 11
\SetRowColor{white}
Cotangent Supplementary & \seqsplit{cot(θ)=-cot(πn-θ)} \tn 
% Row Count 23 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{n ∈ ℕ\textasciitilde{}1\textasciitilde{} = \{1,2,3,4,5,...\}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.7165 cm} x{1.7165 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Periodicity Properties}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Periodicity & \seqsplit{sin(θ)=sin(θ±2πn)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Periodicity & \seqsplit{cos(θ)=cos(θ±2πn)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Periodicity & \seqsplit{tan(θ)=tan(θ±πn)} \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
Cosecant Periodicity & \seqsplit{csc(θ)=csc(θ±2πn)} \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Secant Periodicity & \seqsplit{sec(θ)=sec(θ±2πn)} \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
Cotangent Periodicity & \seqsplit{cot(θ)=cot(θ±πn)} \tn 
% Row Count 11 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{n ∈ ℕ\textasciitilde{}1\textasciitilde{} = \{1,2,3,4,5,...\}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.47619 cm} x{1.95681 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular Parity Properties}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Odd & sin(-θ)=-sin(θ) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Cosine Even & cos(-θ)=cos(θ) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Tangent Odd & tan(-θ)=-tan(θ) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
Cosecant Odd & csc(-θ)=-csc(θ) \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Secant Even & sec(-θ)=sec(θ) \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
Cotangent Odd & cot(-θ)=-cot(θ) \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.95681 cm} x{1.47619 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular Pythagorean Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine-Cosine Pythagorean & sin\textasciicircum{}2\textasciicircum{}(θ)+cos\textasciicircum{}2\textasciicircum{}(θ)=1 \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Secant-Tangent Pythagorean & tan\textasciicircum{}2\textasciicircum{}(θ)+1=sec\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Cosecant-Cotangent Pythagorean & 1+cot\textasciicircum{}2\textasciicircum{}(θ)=csc\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{The last two Pythagorean Identities are obtained by dividing all the terms of the original Sine-Cosine Identity by cos$^{\textrm{2}}$(θ) and sin$^{\textrm{2}}$(θ), respectively}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.20155 cm} x{2.23145 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(C) Half/Multiple-Angle Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Half-Angle & sin(θ/2)=±{\bf{√}}($\frac{1}{2}$(1-cos(θ))) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Half-Angle & cos(θ/2)=±{\bf{√}}($\frac{1}{2}$(1+cos(θ))) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Half-Angle 1 & tan(θ/2)=±{\bf{√}}((1-cos(θ))/(1+cos(θ))) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Tangent Half-Angle 2 & \seqsplit{tan(θ/2)=(1-cos(θ))/sin(θ)} \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Tangent Half-Angle 3 & \seqsplit{tan(θ/2)=sin(θ)/(1+cos(θ))} \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 10 (+ 0)
% Row 6
\SetRowColor{LightBackground}
Sine Double-Angle 1 & sin(2θ)=2sin(θ)cos(θ) \tn 
% Row Count 12 (+ 2)
% Row 7
\SetRowColor{white}
Sine Double-Angle 2 & sin(2θ)=2tan(θ)/(1+tan\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 14 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Cosine Double-Angle 1 & cos(2θ)=cos\textasciicircum{}2\textasciicircum{}(θ)-sin\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 16 (+ 2)
% Row 9
\SetRowColor{white}
Cosine Double-Angle 2 & cos(2θ)=2cos\textasciicircum{}2\textasciicircum{}(θ)-1 \tn 
% Row Count 18 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Cosine Double-Angle 3 & cos(2θ)=1-2sin\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 20 (+ 2)
% Row 11
\SetRowColor{white}
Cosine Double-Angle 4 & cos(2θ)=(1-tan\textasciicircum{}2\textasciicircum{}(θ))/(1+tan\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 22 (+ 2)
% Row 12
\SetRowColor{LightBackground}
Tangent Double-Angle 1 & tan(2θ)=2tan(θ)/(1-tan\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 24 (+ 2)
% Row 13
\SetRowColor{white}
Tangent Double-Angle 2 & \seqsplit{tan(2θ)=2/(cot(θ)-tan(θ))} \tn 
% Row Count 26 (+ 2)
% Row 14
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 26 (+ 0)
% Row 15
\SetRowColor{white}
Sine Triple-Angle & sin(3θ)=3sin(θ)-4sin\textasciicircum{}3\textasciicircum{}(θ) \tn 
% Row Count 28 (+ 2)
% Row 16
\SetRowColor{LightBackground}
Cosine Triple-Angle & cos(3θ)=4cos\textasciicircum{}3\textasciicircum{}(θ)-3cos(θ) \tn 
% Row Count 30 (+ 2)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{3.833cm}{x{1.20155 cm} x{2.23145 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(C) Half/Multiple-Angle Identities (cont)}}  \tn
% Row 17
\SetRowColor{LightBackground}
Tangent Triple-Angle & tan(3θ)=(3tan(θ)-tan\textasciicircum{}3\textasciicircum{}(θ))/(1-3tan\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 2 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Sine Multiple-Angle Formula: sin(nθ)=∑\textasciicircum{}n\textasciicircum{}\textasciitilde{}k=0\textasciitilde{} (\textasciicircum{}n\textasciicircum{}\textasciitilde{}k\textasciitilde{})cos\textasciicircum{}k\textasciicircum{}(θ)sin\textasciicircum{}n-k\textasciicircum{}(θ)sin((π/2)(n-k)) \newline Cosine Multiple-Angle Formula: cos(nθ)=∑\textasciicircum{}n\textasciicircum{}\textasciitilde{}k=0\textasciitilde{} (\textasciicircum{}n\textasciicircum{}\textasciitilde{}k\textasciitilde{})cos\textasciicircum{}k\textasciicircum{}(θ)sin\textasciicircum{}n-k\textasciicircum{}(θ)cos((π/2)(n-k)) \newline All the following identities are true for values that do not cause division by zero}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{0.94023 cm} p{0.6066 cm} x{1.48617 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{Circular Sum/Difference/Product Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine \seqsplit{Sum/Difference} & \seqsplit{sin(θ±φ)} & \seqsplit{sin(θ)cos(φ)±cos(θ)sin(φ)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Sine Sum-Product & \seqsplit{sin(θ)±sin(φ)} & \seqsplit{2sin((θ±φ)/2)cos((θ∓φ)/2)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Sine Product-Sum & \seqsplit{sin(θ)sin(φ)} & $\frac{1}{2}$(cos(θ-φ)-cos(θ+φ)) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Cosine \seqsplit{Sum/Difference} & \seqsplit{cos(θ±φ)} & \seqsplit{cos(θ)cos(φ)∓sin(θ)sin(φ)} \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Cosine Sum-Product & \seqsplit{cos(θ)±cos(φ)} & \seqsplit{2cos((θ±φ)/2)cos((θ∓φ)/2)} \tn 
% Row Count 11 (+ 2)
% Row 5
\SetRowColor{white}
Cosine Product-Sum & \seqsplit{cos(θ)cos(φ)} & $\frac{1}{2}$(cos(θ-φ)+cos(θ+φ)) \tn 
% Row Count 13 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Sine-Cosine Product-Sum & \seqsplit{sin(θ)cos(φ)} & $\frac{1}{2}$(sin(θ-φ)+sin(θ+φ)) \tn 
% Row Count 15 (+ 2)
% Row 7
\SetRowColor{white}
Tangent \seqsplit{Sum/Difference} & \seqsplit{tan(θ±φ)} & \seqsplit{(tan(θ)±tan(φ))/(1∓tan(θ)tan(φ))} \tn 
% Row Count 18 (+ 3)
% Row 8
\SetRowColor{LightBackground}
Tangent Sum & \seqsplit{tan(θ)±tan(φ)} & \seqsplit{sin(θ±φ)/(cos(θ)cos(φ))} \tn 
% Row Count 20 (+ 2)
% Row 9
\SetRowColor{white}
Tangent Product & \seqsplit{tan(θ)tan(φ)} & \seqsplit{(tan(θ)+tan(φ))/(cot(θ)+cot(φ))} \tn 
% Row Count 22 (+ 2)
% Row 10
\SetRowColor{LightBackground}
\seqsplit{Tangent-Cotangent} Product & \seqsplit{tan(θ)cot(φ)} & \seqsplit{(tan(θ)+cot(φ))/(cot(θ)+tan(φ))} \tn 
% Row Count 25 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sine and Cosine Unit Circle}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/crossant_1723925416_sinecosineunitcircle.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{x\textasciicircum{}2\textasciicircum{}+y\textasciicircum{}2\textasciicircum{}=1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.44186 cm} x{1.99114 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular-Inverse Reciprocal Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Reciprocal & arcsin(1/x)=arccsc(x) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Cosine Reciprocal & arccos(1/x)=arcsec(x) \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Reciprocal 1 & arctan(1/x)=arccot(x), {\emph{x\textgreater{}0}} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
Tangent Reciprocal 2 & \seqsplit{arctan(1/x)=arccot(x)-π}, {\emph{x\textless{}0}} \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Cosecant Reciprocal & arccsc(1/x)=arcsin(x) \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
Secant Reciprocal & arcsec(1/x)=arccos(x) \tn 
% Row Count 11 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Cotangent Reciprocal 1 & arccot(1/x)=arctan(x), {\emph{x\textgreater{}0}} \tn 
% Row Count 13 (+ 2)
% Row 7
\SetRowColor{white}
Cotangent Reciprocal 2 & \seqsplit{arccot(1/x)=arctan(x)+π}, {\emph{x\textless{}0}} \tn 
% Row Count 15 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.68217 cm} x{1.75083 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular-Inverse Complimentary Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Complimentary & \seqsplit{arcsin(x)=π/2-arccos(x)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Complimentary & \seqsplit{arccos(x)=π/2-arcsin(x)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Complimentary & \seqsplit{arctan(x)=π/2-arccot(x)} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Cosecant Complimentary & \seqsplit{arccsc(x)=π/2-arcsec(x)} \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Secant Complimentary & \seqsplit{arcsec(x)=π/2-arccsc(x)} \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
Cotangent Complimentary & \seqsplit{arccot(x)=π/2-arctan(x)} \tn 
% Row Count 12 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.64784 cm} x{1.78516 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular-Inverse Negative Input Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Odd & \seqsplit{arcsin(-x)=-arcsin(x)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Translation & \seqsplit{arccos(-x)=π-arccos(x)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Odd & \seqsplit{arctan(-x)=-arctan(x)} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Cosecant Odd & \seqsplit{arccsc(-x)=-arccsc(x)} \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Secant Translation & \seqsplit{arcsec(-x)=π-arcsec(x)} \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
Cotangent Translation & \seqsplit{arccot(-x)=π-arccot(x)} \tn 
% Row Count 12 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.44186 cm} x{1.99114 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(CI) Half/Multiple Substitution Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Half Sine Substitution 1 & $\frac{1}{2}$arcsin(x)=arcsin({\bf{√}}(1+x)/2))-π/4 \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Half Sine Substitution 2 & $\frac{1}{2}$arcsin(x)=π/4-arcsin({\bf{√}}(1-x)/2)) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Half Cosine Substitution 1 & $\frac{1}{2}$arccos(x)=arccos({\bf{√}}((1+x)/2)) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Half Cosine Substitution 2 & $\frac{1}{2}$arccos(x)=π/2-arccos({\bf{√}}(1-x)/2)) \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 8 (+ 0)
% Row 5
\SetRowColor{white}
Double Sine Substitution & 2arcsin(x)=arcsin(2x{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{})), {\emph{|x|≤π/2}} \tn 
% Row Count 11 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Double Cosine Substitution 1 & 2arccos(x)=arccos(2x\textasciicircum{}2\textasciicircum{}-1), {\emph{x≥0}} \tn 
% Row Count 13 (+ 2)
% Row 7
\SetRowColor{white}
Double Cosine Substitution 2 & 2arccos(x)=2π-arccos(2x\textasciicircum{}2\textasciicircum{}-1), {\emph{x≤0}} \tn 
% Row Count 15 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Double Tangent Substitution 1 & 2arctan(x)=arcsin(2x/(1+x\textasciicircum{}2\textasciicircum{})), {\emph{|x|≤1}} \tn 
% Row Count 17 (+ 2)
% Row 9
\SetRowColor{white}
Double Tangent Substitution 2 & 2arctan(x)=±arccos((1-x\textasciicircum{}2\textasciicircum{})/(1+x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 19 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Double Tangent Substitution 3 & 2arctan(x)=arctan(2x/(1-x\textasciicircum{}2\textasciicircum{})), {\emph{|x|\textless{}1}} \tn 
% Row Count 21 (+ 2)
% Row 11
\SetRowColor{white}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 21 (+ 0)
% Row 12
\SetRowColor{LightBackground}
Triple Sine Substitution 1 & 3arcsin(x)=arcsin(3x-4x\textasciicircum{}3\textasciicircum{}), {\emph{|x|≤$\frac{1}{2}$}} \tn 
% Row Count 23 (+ 2)
% Row 13
\SetRowColor{white}
Triple Sine Substitution 2 & 3arcsin(x)=arcsin(4x\textasciicircum{}3\textasciicircum{}-3x)±π, {\emph{|x|≥$\frac{1}{2}$}} \tn 
% Row Count 25 (+ 2)
% Row 14
\SetRowColor{LightBackground}
Triple Cosine Substitution 1 & 3arccos(x)=arccos(3x-4x\textasciicircum{}3\textasciicircum{}), {\emph{|x|≤$\frac{1}{2}$}} \tn 
% Row Count 27 (+ 2)
% Row 15
\SetRowColor{white}
Triple Cosine Substitution 2 & 3arccos(x)=arccos(4x\textasciicircum{}3\textasciicircum{}-3x)+π±π, {\emph{|x|≥$\frac{1}{2}$}} \tn 
% Row Count 29 (+ 2)
% Row 16
\SetRowColor{LightBackground}
Triple Tangent Substitution 1 & 3arctan(x)=arctan((3x-x\textasciicircum{}3\textasciicircum{})/(1-3x\textasciicircum{}2\textasciicircum{})), {\emph{|x|≤√3/3}} \tn 
% Row Count 32 (+ 3)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{3.833cm}{x{1.44186 cm} x{1.99114 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(CI) Half/Multiple Substitution Identities (cont)}}  \tn
% Row 17
\SetRowColor{LightBackground}
Triple Tangent Substitution 2 & 3arctan(x)=arctan((3x-x\textasciicircum{}3\textasciicircum{})/(1-3x\textasciicircum{}2\textasciicircum{}))±π, {\emph{|x|≥√3/3}} \tn 
% Row Count 3 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.33887 cm} x{2.09413 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Circular-Inverse Sum/Difference Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Sum/Difference & arcsin(x)±arcsin(y)=arcsin(x{\bf{√}}(1-y\textasciicircum{}2\textasciicircum{})±y{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Cosine Sum/Difference & arccos(x)±arccos(y)=arccos(xy∓{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}){\bf{√}}(1-y\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Cosine-Sine Sum/Difference & arccos(x)±arcsin(y)=arccos(x{\bf{√}}(1-y\textasciicircum{}2\textasciicircum{})∓y{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
Tangent Sum/Difference & \seqsplit{arctan(x)±arctan(y)=arctan((x±y)/(1∓xy))}, {\emph{1∓xy≠0}} \tn 
% Row Count 12 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.0299 cm} x{2.4031 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Law of Sines/Cosines/Tangents}}  \tn
% Row 0
\SetRowColor{LightBackground}
Law of Sines 1 & \seqsplit{sin(α)/a=sin(β)/b=sin(γ)/c} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Law of Sines 2 & \seqsplit{a/sin(α)=b/sin(β)=c/sin(γ)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Law of Cosines 1 & a\textasciicircum{}2\textasciicircum{}=b\textasciicircum{}2\textasciicircum{}+c\textasciicircum{}2\textasciicircum{}-2bccos(α) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Law of Cosines 2 & b\textasciicircum{}2\textasciicircum{}=a\textasciicircum{}2\textasciicircum{}+c\textasciicircum{}2\textasciicircum{}-2accos(β) \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Law of Cosines 3 & c\textasciicircum{}2\textasciicircum{}=a\textasciicircum{}2\textasciicircum{}+b\textasciicircum{}2\textasciicircum{}-2abcos(γ) \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
Law of Tangents 1 & \seqsplit{(a-b)/(a+b)=tan((α-β)/2)/tan((α+β)/2)} \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Law of Tangents 2 & \seqsplit{(b-c)/(b+c)=tan((β-γ)/2)/tan((β+γ)/2)} \tn 
% Row Count 14 (+ 2)
% Row 7
\SetRowColor{white}
Law of Tangents 3 & \seqsplit{(c-a)/(c+a)=tan((γ-α)/2)/tan((γ+α)/2)} \tn 
% Row Count 16 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Side lengths a, b, and c are opposite of the angles α, β, and γ, respectively.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.61351 cm} x{1.81949 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Measurements And Formulas}}  \tn
% Row 0
\SetRowColor{LightBackground}
Radians-Degrees & 1 radian=180/π degrees; 1=(180/π)° \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Degrees-Radians & 1 degree=π/180 radians; 1°=π/180 radians \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Degrees, Minutes, and Seconds (DMS) & 1 degree=60 minutes=3600 \seqsplit{seconds;1°=60'=3600''} \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
Arc Length/Angular Displacement & s=rθ units \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Sector Area & $\frac{1}{2}$r\textasciicircum{}2\textasciicircum{}θ units\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 11 (+ 1)
% Row 5
\SetRowColor{white}
Area of a Triangle & A\textasciitilde{}T\textasciitilde{}=$\frac{1}{2}$bh units\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 12 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Area of a Circle & A\textasciitilde{}C\textasciitilde{}=πr\textasciicircum{}2\textasciicircum{} units\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 13 (+ 1)
% Row 7
\SetRowColor{white}
Pythagorean Theorem & a\textasciicircum{}2\textasciicircum{}+b\textasciicircum{}2\textasciicircum{}=c\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 15 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Radians are unitless \newline a, b, and c are side lengths of a right-triangle}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Tangent Unit Circle}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/crossant_1723925842_tangentunitcircle.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.27021 cm} x{2.16279 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(H) Quotient \& Reciprocal Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Tangent Quotient & \seqsplit{tanh(θ)=sinh(θ)/cosh(θ)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cotangent Quotient & \seqsplit{coth(θ)=cosh(θ)/sinh(θ)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 4 (+ 0)
% Row 3
\SetRowColor{white}
Sine Reciprocal & sinh(θ)=1/csch(θ) \tn 
% Row Count 6 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Cosine Reciprocal & cosh(θ)=1/sech(θ) \tn 
% Row Count 8 (+ 2)
% Row 5
\SetRowColor{white}
Tangent Reciprocal & tanh(θ)=1/coth(θ) \tn 
% Row Count 10 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Cosecant Reciprocal & csch(θ)=1/sinh(θ) \tn 
% Row Count 12 (+ 2)
% Row 7
\SetRowColor{white}
Secant Reciprocal & sech(θ)=1/cosh(θ) \tn 
% Row Count 14 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Cotangent Reciprocal & coth(θ)=1/tanh(θ) \tn 
% Row Count 16 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{All the following identities are true for values that do not cause division by zero}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.40753 cm} x{2.02547 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Hyperbolic Parity Properties}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Odd & sinh(-θ)=-sinh(θ) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Cosine Even & cosh(-θ)=cosh(θ) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Tangent Odd & tanh(-θ)=-tanh(θ) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
Cosecant Odd & csch(-θ)=-csch(θ) \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Secant Even & sech(-θ)=sech(θ) \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
Cotangent Odd & coth(-θ)=-coth(θ) \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.45584 cm} x{1.27386 cm} p{0.3033 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{Hyperbolic Pythagorean Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine-Cosine Pythagorean & cosh\textasciicircum{}2\textasciicircum{}(θ)-sinh\textasciicircum{}2\textasciicircum{}(θ)=1 &  \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Secant Pythagorean & 1-tanh\textasciicircum{}2\textasciicircum{}(θ)=sech\textasciicircum{}2\textasciicircum{}(θ) &  \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Cosecant Pythagorean & coth\textasciicircum{}2\textasciicircum{}(θ)-1=csch\textasciicircum{}2\textasciicircum{}(θ) &  \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{3.833cm}}{The last two Hyperbolic Pythagorean Identities are obtained by dividing all the terms of the original Sine-Cosine Identity by cosh$^{\textrm{2}}$(θ) and sinh$^{\textrm{2}}$(θ), respectively}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.20155 cm} x{2.23145 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(H) Half-Angle \& Multiple-Angle Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Half-Angle & sinh(θ/2)=±{\bf{√}}($\frac{1}{2}$(cosh(θ)-1)) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Half-Angle & cosh(θ/2)={\bf{√}}($\frac{1}{2}$(cosh(θ)+1)) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Half-Angle 1 & tanh(θ/2)=±{\bf{√}}((cosh(θ)-1)/(cosh(θ)+1)) \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
Tangent Half-Angle 2 & \seqsplit{tanh(θ/2)=(cosh(θ)-1)/sinh(θ)} \tn 
% Row Count 9 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Tangent Half-Angle 3 & \seqsplit{tanh(θ/2)=sinh(θ)/(cosh(θ)+1)} \tn 
% Row Count 11 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 11 (+ 0)
% Row 6
\SetRowColor{LightBackground}
Sine Double-Angle 1 & \seqsplit{sinh(2θ)=2sinh(θ)cosh(θ)} \tn 
% Row Count 13 (+ 2)
% Row 7
\SetRowColor{white}
Sine Double-Angle 2 & sinh(2θ)=2tanh(θ)/(1-tanh\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 15 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Cosine Double-Angle 1 & cosh(2θ)=cosh\textasciicircum{}2\textasciicircum{}(θ)+sinh\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 17 (+ 2)
% Row 9
\SetRowColor{white}
Cosine Double-Angle 2 & cosh(2θ)=2cosh\textasciicircum{}2\textasciicircum{}(θ)-1 \tn 
% Row Count 19 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Cosine Double-Angle 3 & cosh(2θ)=1+2sinh\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 21 (+ 2)
% Row 11
\SetRowColor{white}
Cosine Double-Angle 4 & cosh(2θ)=(1+tanh\textasciicircum{}2\textasciicircum{}(θ))/(1-tanh\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 23 (+ 2)
% Row 12
\SetRowColor{LightBackground}
Tangent Double Angle 1 & tanh(2θ)=2tanh(θ)/(1+tanh\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 25 (+ 2)
% Row 13
\SetRowColor{white}
Tangent Double Angle 2 & \seqsplit{tanh(2θ)=2/(coth(θ)+tanh(θ))} \tn 
% Row Count 27 (+ 2)
% Row 14
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 27 (+ 0)
% Row 15
\SetRowColor{white}
Sine Triple-Angle & sinh(3θ)=3sinh(θ)+4sinh\textasciicircum{}3\textasciicircum{}(θ) \tn 
% Row Count 29 (+ 2)
% Row 16
\SetRowColor{LightBackground}
Cosine Triple-Angle & cosh(3θ)=4cosh\textasciicircum{}3\textasciicircum{}(θ)-3cosh(θ) \tn 
% Row Count 31 (+ 2)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{3.833cm}{x{1.20155 cm} x{2.23145 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(H) Half-Angle \& Multiple-Angle Identities (cont)}}  \tn
% Row 17
\SetRowColor{LightBackground}
Tangent Triple-Angle & tanh(3θ)=(3tan(θ)+tan\textasciicircum{}3\textasciicircum{}(θ))/(1+3tan\textasciicircum{}2\textasciicircum{}(θ)) \tn 
% Row Count 2 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.03122 cm} p{0.66726 cm} x{1.33452 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{(H) Sum/Difference/Product Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine \seqsplit{Sum/Difference} & \seqsplit{sinh(θ±φ)} & \seqsplit{sinh(θ)cosh(φ)±cosh(θ)sinh(φ)} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Sine Sum-Product & \seqsplit{sinh(θ)±sinh(φ)} & \seqsplit{2sinh((θ±φ)/2)cosh((θ∓φ)/2)} \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Sine Product-Sum & \seqsplit{sinh(θ)sinh(φ)} & $\frac{1}{2}$(cosh(θ+φ)-cosh(θ-φ)) \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
Cosine \seqsplit{Sum/Difference} & \seqsplit{cosh(θ±φ)} & \seqsplit{cosh(θ)cosh(φ)∓sinh(θ)sinh(φ)} \tn 
% Row Count 13 (+ 4)
% Row 4
\SetRowColor{LightBackground}
Cosine Sum-Product 1 & \seqsplit{cosh(θ)+cosh(φ)} & \seqsplit{2cosh((θ+φ)/2)cosh((θ-φ)/2)} \tn 
% Row Count 16 (+ 3)
% Row 5
\SetRowColor{white}
Cosine Sum-Product 2 & \seqsplit{cosh(θ)-cosh(φ)} & \seqsplit{2sinh((θ+φ)/2)sinh((θ-φ)/2)} \tn 
% Row Count 19 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Cosine Product-Sum & \seqsplit{cosh(θ)cosh(φ)} & $\frac{1}{2}$(cosh(θ+φ)+cosh(θ-φ)) \tn 
% Row Count 22 (+ 3)
% Row 7
\SetRowColor{white}
Sine-Cosine Product-Sum & \seqsplit{sinh(θ)cosh(φ)} & $\frac{1}{2}$(sinh(θ+φ)+sinh(θ-φ)) \tn 
% Row Count 25 (+ 3)
% Row 8
\SetRowColor{LightBackground}
Tangent \seqsplit{Sum/Difference} & \seqsplit{tanh(θ±φ)} & \seqsplit{(tanh(θ)±tanh(φ))/(1±tanh(θ)tanh(φ))} \tn 
% Row Count 29 (+ 4)
% Row 9
\SetRowColor{white}
Tangent Sum & \seqsplit{tanh(θ)±tanh(φ)} & \seqsplit{sinh(θ±φ)/(cosh(θ)cosh(φ))} \tn 
% Row Count 32 (+ 3)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{3.833cm}{x{1.03122 cm} p{0.66726 cm} x{1.33452 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{(H) Sum/Difference/Product Identities (cont)}}  \tn
% Row 10
\SetRowColor{LightBackground}
Tangent Product & \seqsplit{tanh(θ)tanh(φ)} & \seqsplit{(tanh(θ)+tanh(φ))/(coth(θ)+coth(φ))} \tn 
% Row Count 4 (+ 4)
% Row 11
\SetRowColor{white}
\seqsplit{Tangent-Cotangent} Product & \seqsplit{tanh(θ)coth(φ)} & \seqsplit{(tanh(θ)+coth(φ))/(coth(θ)+tanh(φ))} \tn 
% Row Count 8 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Right-Triangle Relations}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/crossant_1723925620_righttriangletrigrelations.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.61351 cm} x{1.81949 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Hyperbolic-Inverse Reciprocal Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Reciprocal & \seqsplit{arcsinh(1/x)=arccsch(x)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Reciprocal & \seqsplit{arccosh(1/x)=arcsech(x)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Reciprocal & \seqsplit{arctanh(1/x)=arccoth(x)} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Cosecant Reciprocal & \seqsplit{arccsch(1/x)=arcsinh(x)} \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Secant Reciprocal & \seqsplit{arcsech(1/x)=arccosh(x)} \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
Cotangent Reciprocal & \seqsplit{arccoth(1/x)=arctanh(x)} \tn 
% Row Count 12 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.64784 cm} x{1.78516 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(HI) Negative Input Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Inverse Sine Odd & \seqsplit{arcsinh(-x)=-arcsinh(x)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Inverse Tangent Odd & \seqsplit{arctanh(-x)=-arctanh(x)} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Inverse Cosecant Odd & \seqsplit{arccsch(-x)=-arccsch(x)} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Inverse Cotangent Odd & \seqsplit{arccoth(-x)=-arccoth(x)} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.3732 cm} x{2.0598 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(HI) Half/Multiple Substitution Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Half Sine Substitution & $\frac{1}{2}$arcsinh(x)=±arcsinh({\bf{√}}({\bf{√}}((1+x\textasciicircum{}2\textasciicircum{})-1)/2)) \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Half Cosine Substitution & $\frac{1}{2}$arccosh(x)=arccosh({\bf{√}}((x+1)/2)) \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Half Tangent Substitution & $\frac{1}{2}$arctanh(x)=arctanh(x/(1+{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{}))) \tn 
% Row Count 7 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 7 (+ 0)
% Row 4
\SetRowColor{LightBackground}
Double Sine Substitution 1 & 2arcsinh(x)=arcsinh(2x{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
Double Sine Substitution 2 & 2arcsinh(x)=±arccosh(2x\textasciicircum{}2\textasciicircum{}+1) \tn 
% Row Count 11 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Double Cosine Substitution & 2arccosh(x)=arccosh(2x\textasciicircum{}2\textasciicircum{}-1), {\emph{x≥1}} \tn 
% Row Count 13 (+ 2)
% Row 7
\SetRowColor{white}
Double Tangent Substitution & 2arctanh(x)=arctanh(2x/(1+x\textasciicircum{}2\textasciicircum{})), {\emph{|x|\textless{}1}} \tn 
% Row Count 15 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 15 (+ 0)
% Row 9
\SetRowColor{white}
Triple Sine Substitution & 3arcsinh(x)=arcsinh(3x+4x\textasciicircum{}3\textasciicircum{}) \tn 
% Row Count 17 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Triple Cosine Substitution & 3arccosh(x)=arccosh(4x\textasciicircum{}3\textasciicircum{}-3x) \tn 
% Row Count 19 (+ 2)
% Row 11
\SetRowColor{white}
Triple Tangent Substitution & 3arctanh(x)=arctanh((3x+x\textasciicircum{}3\textasciicircum{})/(1+3x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 21 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.40753 cm} x{2.02547 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(HI) Sum/Difference Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Sum/Difference & arcsinh(x)±arcsinh(y)=arcsinh(x{\bf{√}}(y\textasciicircum{}2\textasciicircum{}+1)±y{\bf{√}}(x\textasciicircum{}2\textasciicircum{}+1)) \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Cosine Sum/Difference & arccosh(x)±arccosh(y)=arccosh(xy±{\bf{√}}((x\textasciicircum{}2\textasciicircum{}-1)(y\textasciicircum{}2\textasciicircum{}-1))) \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Sine-Cosine Sum/Difference 1 & arcsinh(x)±arccosh(y)=arcsinh(xy±{\bf{√}}((x\textasciicircum{}2\textasciicircum{}+1)(y\textasciicircum{}2\textasciicircum{}-1)) \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
Sine-Cosine Sum/Difference 2 & arcsinh(x)±arccosh(y)=±arccosh(y{\bf{√}}(x\textasciicircum{}2\textasciicircum{}+1)±x{\bf{√}}(y\textasciicircum{}2\textasciicircum{}-1)) \tn 
% Row Count 12 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Tangent Sum/Difference & \seqsplit{arctanh(x)±arctanh(y)=arctanh((x±y)/(1±xy))} \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.44186 cm} x{1.99114 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{(HI) \seqsplit{Logarithmic/Compositional} Conversions}}  \tn
% Row 0
\SetRowColor{LightBackground}
Sine Logarithmic & ln(x)=arcsinh((x\textasciicircum{}2\textasciicircum{}-1)/2x), {\emph{x\textgreater{}0}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cosine Logarithmic & ln(x)=±arccosh((x\textasciicircum{}2\textasciicircum{}+1)/2x) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Tangent Logarithmic & ln(x)=arctanh((x\textasciicircum{}2\textasciicircum{}-1)/(x\textasciicircum{}2\textasciicircum{}+1)) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 6 (+ 0)
% Row 4
\SetRowColor{LightBackground}
Sine \seqsplit{Hyperbolic-Circular} 1 & \seqsplit{arcsinh(tan(x))=ln(sec(x+πn)+tan(x+πn))} \tn 
% Row Count 8 (+ 2)
% Row 5
\SetRowColor{white}
Sine \seqsplit{Hyperbolic-Circular} 2 & \seqsplit{arcsinh(tan(x))=±arccosh(sec(x+πn))} \tn 
% Row Count 10 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Tangent \seqsplit{Hyperbolic-Circular} 1 & \seqsplit{arctanh(cos(2x))=ln(|cot(x)|)} \tn 
% Row Count 12 (+ 2)
% Row 7
\SetRowColor{white}
Tangent \seqsplit{Hyperbolic-Circular} 2 & \seqsplit{±arctanh(sin(x))=±arcsinh(tan(x))} \tn 
% Row Count 14 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 14 (+ 0)
% Row 9
\SetRowColor{white}
Sine Cofunctional 1 & arcsinh(x)=arctanh(x/{\bf{√}}(1+x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 16 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Sine Cofunctional 2 & arcsinh(x)=±arccosh({\bf{√}}(1+x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 18 (+ 2)
% Row 11
\SetRowColor{white}
Cosine Cofunctional 1 & arccosh(x)=|arcsinh({\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1))|, {\emph{x≥1}} \tn 
% Row Count 20 (+ 2)
% Row 12
\SetRowColor{LightBackground}
Cosine Cofunctional 2 & arccosh(x)=|arctanh({\bf{√}}(x\textasciicircum{}2\textasciicircum{}-1)/x)|, {\emph{x≥1}} \tn 
% Row Count 23 (+ 3)
% Row 13
\SetRowColor{white}
Tangent Cofunctional 1 & arctanh(x)=arcsinh(x/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 25 (+ 2)
% Row 14
\SetRowColor{LightBackground}
Tangent Cofunctional 2 & arctanh(x)=±arccosh(1/{\bf{√}}(1-x\textasciicircum{}2\textasciicircum{})) \tn 
% Row Count 27 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Unit Hyperbola}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/crossant_1724009404_hyperbolicrelations.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{x\textasciicircum{}2\textasciicircum{}-y\textasciicircum{}2\textasciicircum{}=1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}