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+ </head>
+ <body class="markdown-preview-plus-view">
+ <h1>Circular functions</h1>
+<h2>Radians and degrees</h2>
+<span class="math"><script type="math/tex; mode=display">1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}
+</script></span>
+<h2>Exact values</h2>
+<h2><span class="math"><script type="math/tex">\sin</script></span> and <span class="math"><script type="math/tex">\cos</script></span> graphs</h2>
+<span class="math"><script type="math/tex; mode=display">f(x)=a \sin(bx-c)+d
+</script></span>
+<span class="math"><script type="math/tex; mode=display">f(x)=a \cos(bx-c)+d
+</script></span>
+<p>where<br>
+<span class="math"><script type="math/tex">a</script></span> is the <span class="math"><script type="math/tex">y</script></span>-dilation (amplitude)<br>
+<span class="math"><script type="math/tex">b</script></span> is the <span class="math"><script type="math/tex">x</script></span>-dilation (period)<br>
+<span class="math"><script type="math/tex">c</script></span> is the <span class="math"><script type="math/tex">x</script></span>-shift (phase)<br>
+<span class="math"><script type="math/tex">d</script></span> is the <span class="math"><script type="math/tex">y</script></span>-shift (equilibrium position)</p>
+<p>Domain is <span class="math"><script type="math/tex">\mathbb{R}</script></span><br>
+Range is <span class="math"><script type="math/tex">[-b+c, b+c]</script></span>;</p>
+<p>Graph of <span class="math"><script type="math/tex">\cos(x)</script></span> starts at <span class="math"><script type="math/tex">(0,1)</script></span>. Graph of <span class="math"><script type="math/tex">\sin(x)</script></span> starts at <span class="math"><script type="math/tex">(0,0)</script></span>.</p>
+<p><strong>Mean / equilibrium:</strong> line that the graph oscillates around (<span class="math"><script type="math/tex">y=d</script></span>)</p>
+<h2>Solving trig equations</h2>
+<ol>
+<li>Solve domain for <span class="math"><script type="math/tex">n\theta</script></span></li>
+<li>Find solutions for <span class="math"><script type="math/tex">n\theta</script></span></li>
+<li>Divide solutions by <span class="math"><script type="math/tex">n</script></span></li>
+</ol>
+<p><span class="math"><script type="math/tex">\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])</script></span><br>
+<span class="math"><script type="math/tex">2\theta=\sin^{-1}{\sqrt{3} \over 2}</script></span><br>
+<span class="math"><script type="math/tex">2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}</script></span><br>
+<span class="math"><script type="math/tex">\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}</script></span></p>
+<h3>Amplitude</h3>
+<p>Amplitude of <span class="math"><script type="math/tex">a</script></span> means graph oscillates between <span class="math"><script type="math/tex">+a</script></span> and <span class="math"><script type="math/tex">-a</script></span> in <span class="math"><script type="math/tex">y</script></span>-axis</p>
+<p><span class="math"><script type="math/tex">a=0</script></span> produces straight line<br>
+<span class="math"><script type="math/tex">a\lt0</script></span> inverts the phase (<span class="math"><script type="math/tex">\sin</script></span> becomes <span class="math"><script type="math/tex">\cos</script></span>, vice vera)</p>
+<h3>Period</h3>
+<p>Period <span class="math"><script type="math/tex">T</script></span> is <span class="math"><script type="math/tex">{2 \pi}\over b</script></span><br>
+<span class="math"><script type="math/tex">b=0</script></span> produces straight line<br>
+<span class="math"><script type="math/tex">b\lt0</script></span> inverts the phase</p>
+<h3>Phase</h3>
+<p><span class="math"><script type="math/tex">c</script></span> moves the graph left-right in the <span class="math"><script type="math/tex">x</script></span> axis.<br>
+If <span class="math"><script type="math/tex">c=T={{2\pi}\over b}</script></span>, the graph has no actual phase shift.</p>
+<h2>Symmetry</h2>
+<span class="math"><script type="math/tex; mode=display">\sin(\theta+{\pi\over 2})=\sin\theta
+</script></span>
+<span class="math"><script type="math/tex; mode=display">\sin(\theta+\pi)=-\sin\theta
+</script></span>
+<span class="math"><script type="math/tex; mode=display">\cos(\theta+{\pi \over 2})=-\cos\theta
+</script></span>
+<span class="math"><script type="math/tex; mode=display">\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)
+</script></span>
+<h2>Pythagorean identity</h2>
+<span class="math"><script type="math/tex; mode=display">\cos^2\theta+\sin^2\theta=1
+</script></span>
+<h2>Complementary relationships</h2>
+<span class="math"><script type="math/tex; mode=display">\sin({\pi \over 2} - \theta)=\cos\theta
+</script></span>
+<span class="math"><script type="math/tex; mode=display">\cos({\pi \over 2} - \theta)=\sin\theta
+</script></span>
+<span class="math"><script type="math/tex; mode=display">\sin\theta=-\cos(\theta+{\pi \over 2})
+</script></span>
+<span class="math"><script type="math/tex; mode=display">\cos\theta=\sin(\theta+{\pi \over 2})
+</script></span>
+<h2><span class="math"><script type="math/tex">\tan</script></span> graph</h2>
+<span class="math"><script type="math/tex; mode=display">y=a\tan(nx)
+</script></span>
+<p>where<br>
+<span class="math"><script type="math/tex">a</script></span> is <span class="math"><script type="math/tex">x</script></span>-dilation (period)<br>
+<span class="math"><script type="math/tex">n</script></span> is <span class="math"><script type="math/tex">y</script></span>-dilation (<span class="math"><script type="math/tex">\equiv</script></span> amplitude)<br>
+period <span class="math"><script type="math/tex">T</script></span> is <span class="math"><script type="math/tex">\pi \over n</script></span><br>
+range is <span class="math"><script type="math/tex">R</script></span><br>
+roots at <span class="math"><script type="math/tex">x={k\pi \over n}</script></span><br>
+asymptotes at <span class="math"><script type="math/tex">x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}</script></span><br>
+<strong>Asymptotes should always have equations and arrow pointing up</strong></p>
+
+ </body>
+</html>
--- /dev/null
+\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
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+%
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+\usepackage{multicol}
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+\makeatother
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+% margins by default, and it is still possible to overwrite the defaults
+% using explicit options in \includegraphics[width, height, ...]{}
+\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
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+
+% set default figure placement to htbp
+\makeatletter
+\def\fps@figure{htbp}
+\makeatother
+
+
+\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}
