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\date{}

\begin{document}

\columnsbegin
\hypertarget{circular-functions}{%
\section{Circular functions}\label{circular-functions}}

\hypertarget{radians-and-degrees}{%
\subsection{Radians and degrees}\label{radians-and-degrees}}

\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]

\hypertarget{exact-values}{%
\subsection{Exact values}\label{exact-values}}

\includegraphics[scale=0.5]{./graphics/exact-values-1.png}

\hypertarget{sin-and-cos-graphs}{%
\subsection{\texorpdfstring{\(\sin\) and \(\cos\)
graphs}{\textbackslash{}sin and \textbackslash{}cos graphs}}\label{sin-and-cos-graphs}}

\[f(x)=a \sin(bx-c)+d\] \[f(x)=a \cos(bx-c)+d\]

where

\begin{itemize}
\tightlist
\item
  \(a\) is the \(y\)-dilation (amplitude)
\item
  \(b\) is the \(x\)-dilation (period)
\item
  \(c\) is the \(x\)-shift (phase)
\item
  \(d\) is the \(y\)-shift (equilibrium position)
\end{itemize}

Domain is \(\mathbb{R}\)

Range is \([-b+c, b+c]\);

Graph of \(\cos(x)\) starts at \((0,1)\). Graph of \(\sin(x)\) starts at
\((0,0)\).

\textbf{Mean / equilibrium:} line that the graph oscillates around
(\(y=d\))

\hypertarget{amplitude}{%
\subsubsection{Amplitude}\label{amplitude}}

Amplitude of \(a\) means graph oscillates between \(+a\) and \(-a\) in
\(y\)-axis

\(a=0\) produces straight line

\(a < 0\) inverts the phase (\(\sin\) becomes \(\cos\), vice vera)

\hypertarget{period}{%
\subsubsection{Period}\label{period}}

Period \(T\) is \({2 \pi}\over b\)

\(b=0\) produces straight line

\(b<0\) inverts the phase

\hypertarget{phase}{%
\subsubsection{Phase}\label{phase}}

\(c\) moves the graph left-right in the \(x\) axis.

If \(c=T={{2\pi}\over b}\), the graph has no actual phase shift.

\hypertarget{symmetry}{%
\subsection{Symmetry}\label{symmetry}}

\[\sin(\theta+{\pi\over 2})=\sin\theta\]
\[\sin(\theta+\pi)=-\sin\theta\]

\[\cos(\theta+{\pi \over 2})=-\cos\theta\]
\[\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)\]

\hypertarget{pythagorean-identity}{%
\subsection{Pythagorean identity}\label{pythagorean-identity}}

\[\cos^2\theta+\sin^2\theta=1\]

\hypertarget{complementary-relationships}{%
\subsection{Complementary
relationships}\label{complementary-relationships}}

\[\sin({\pi \over 2} - \theta)=\cos\theta\]
\[\cos({\pi \over 2} - \theta)=\sin\theta\]

\[\sin\theta=-\cos(\theta+{\pi \over 2})\]
\[\cos\theta=\sin(\theta+{\pi \over 2})\]

\hypertarget{tan-graph}{%
\subsection{\texorpdfstring{\(\tan\)
graph}{\textbackslash{}tan graph}}\label{tan-graph}}

\[y=a\tan(nx)\]

where

\begin{itemize}
\tightlist
\item
  \(a\) is \(x\)-dilation (period)
\item
  \(n\) is \(y\)-dilation (\(\equiv\) amplitude)
\item
  period \(T\) is \(\pi \over n\)
\item
  range is \(R\)
\item
  roots at \(x={k\pi \over n}\)
\item
  asymptotes at \(x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}\)
\end{itemize}

\textbf{Asymptotes should always have equations and arrow pointing up}

\hypertarget{solving-trig-equations}{%
\subsection{Solving trig equations}\label{solving-trig-equations}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Solve domain for \(n\theta\)
\item
  Find solutions for \(n\theta\)
\item
  Divide solutions by \(n\)
\end{enumerate}

\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\)

\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\)

\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)

\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)
\columnsend
\end{document}