\section{Circular functions}

\subsection*{Radians and degrees}

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

\subsection*{Exact values}
\adjustbox{trim=0.7cm 0cm}{
  \begin{tikzpicture}
    \matrix{
      \begin{scope}[scale=0.8]
        \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] {\(\dfrac{\pi}{4}\)};
        \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\dfrac{\pi}{4}\)};
        \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{scope}
      &
      \begin{scope}[scale=0.8]
        \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] {\(\dfrac{\pi}{6}\)};
        \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\dfrac{\pi}{3}\)};
        \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{scope}
      \\
    };
  \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*{\(\sin\) 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}

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

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

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

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