\section{Circular functions}

\subsection*{Radians and degrees}

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

\subsection*{Exact values}


    \begin{tikzpicture}[scale=0.75]
      \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)};
      \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\frac{\pi}{4}\)};
      \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\frac{\pi}{4}\)};
      \draw [black] (0,0) coordinate (A) (3,0) coordinate (B) (3,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C};
    \end{tikzpicture}
    \begin{tikzpicture}[scale=0.75]
      \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2};
      \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\frac{\pi}{6}\)};
      \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\frac{\pi}{3}\)};
      \draw [black] (5.19,0) coordinate (A) (0,0) coordinate (B) (0,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C};
    \end{tikzpicture}

                  \subsection*{Compound angle formulas}


                  \begin{align*}
                    \cos(x \pm y) &= \cos x + \cos y \mp \sin x \sin y \\
                    \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \\
                    \tan(x \pm y) &= {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}
                  \end{align*}

                  \subsection*{Double angle formulas}

                  \begin{align*}
                    \cos 2x &= \cos^2 x - \sin^2 x \\
                    & = 1 - 2\sin^2 x \\
                    & = 2 \cos^2 x -1 \\ 
                    \sin 2x &= 2 \sin x \cos x \\
                    \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x}
                  \end{align*}



\subsection*{Symmetry}

\begin{align*}
  \sin(\theta+\frac{\pi}{2}) &= \sin\theta \\
  \sin(\theta+\pi) &= -\sin\theta \\ \\
  \cos(\theta+\frac{\pi}{2}) &= -\cos\theta \\
  \cos(\theta+\pi) &= -\cos(\theta+\frac{3\pi}{2}) \\
  &= \cos(-\theta)
\end{align*}

\subsection*{Complementary relationships}

\begin{align*}
  \sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
  &= -\cos(\theta+\frac{\pi}{2}) \\
  \cos\theta &= \sin(\frac{\pi}{2} - \theta) \\ 
  &= \sin(\theta+\frac{\pi}{2})
\end{align*}

\subsection*{Pythagorean identity}

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

                  \subsection*{Inverse circular functions}

                  \begin{tikzpicture}
                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
                      \addplot[mark=*, blue] coordinates {(1,0)};
                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
                    \end{axis}
                  \end{tikzpicture}\\

                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)

                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)

                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)

                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)

                  \begin{tikzpicture}
                    \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
                      \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
                    \end{axis}
                  \end{tikzpicture}

\subsection*{\(\sin\) and \(\cos\) graphs}

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

where:
\begin{description}
  \item Period \(=\frac{2\pi}{n}\)
  \item dom \(= \mathbb{R}\)
  \item ran \(= [-b+c, b+c]\);
  \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
  \item 0 amplitidue \(\implies\) straight line
  \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
  \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
\end{description}

\subsection*{\(\tan\) graphs}

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

\begin{description}
  \item Period \(= \dfrac{\pi}{n}\)
  \item Range is \(\mathbb{R}\)
  \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
  \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
\end{description}

\textbf{Asymptotes should always have equations}

\subsection*{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}

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

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

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

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