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# Trigonometry

## Right angle triangles

\usetikzlibrary{arrows,positioning, calc}
\tikzset{
%Define standard arrow tip
>=stealth',
% Define arrow style
pil/.style={->,thick}
}

\begin{document}
\begin{tikzpicture}

\coordinate (A) at (0,0);
\coordinate (B) at (2,0);
\coordinate (C) at (2,2);

\coordinate[label=below:adj](c) at ($(A)!.5!(B)$);
\coordinate[label=above left:hyp](b)  at ($(A)!.5!(C)$);
\coordinate[label=right:opp](a) at ($(B)!.5!(C)$);

\draw (1.75,0) -- (1.75,0.25) -- (2,0.25);

\draw (0,0) -- (0:0.85cm) arc (0:45:.85cm);
\draw (0.55cm,0.25cm) node {$\theta$};

\draw [line width=1.5pt] (A) -- (B) -- (C) -- cycle;
\end{tikzpicture}
\end{document}

\begin{align*} \sin \theta &= \frac{\text{opposite}}{\text{hypotenuse}} \\ \cos \theta &= \frac{\text{adjacent}}{\text{hypotenuse}}\\ \tan \theta &= \frac{\sin{\theta}}{\cos{\theta}} = \frac{\text{opposite}}{\text{adjacent}} \end{align*}

## All triangles

\usetikzlibrary{arrows,positioning, calc}
\tikzset{
%Define standard arrow tip
>=stealth',
% Define arrow style
pil/.style={->,thick}
}

\begin{document}
\begin{tikzpicture}

\coordinate[label=left:$A$]  (A) at (0,0);
\coordinate[label=right:$B$] (B) at (4,0);
\coordinate[label=above:$C$] (C) at (2,3.464);

\coordinate[label=below:$c$](c)       at ($(A)!.5!(B)$);
\coordinate[label=left:$b$](b)  at ($(A)!.5!(C)$);
\coordinate[label=right:$a$](a) at ($(B)!.5!(C)$);

% angle alpha
\draw (0,0) -- (0:0.75cm) arc (0:60:.75cm);
\draw (0.35cm,0.25cm) node {$\alpha$};

% angle beta
\begin{scope}[shift={(4cm,0cm)}]
\draw (0,0) -- (-180:0.75cm) arc (180:120:0.75cm);
\draw[color=gray, dashed] (0,0) -- node[sloped, above=-0.1cm] {$\scriptstyle h_b$} (150:3.464cm);
\draw (150:0.5cm) node {$\beta$};
\end{scope}

% angle gamma
\begin{scope}[shift={(60:4)}]
\draw (0,0) -- (-120:.75cm) arc (-120:-60:.75cm);
\draw[color=gray, dashed] (0,0) -- node[right=-0.1cm] {$\scriptstyle h_c$} (-90:3.464cm);
\draw (-90:0.5cm) node {$\gamma$};
\end{scope}

% Height with label
\draw[color=gray, dashed] (0,0) -- node[sloped, above=-0.1cm] {$\scriptstyle h_a$} (30:3.464cm);

% The triangle
\draw [line width=1.5pt] (A) -- (B) -- (C) -- cycle;
\end{tikzpicture}
\end{document}

$\alpha + \beta + \gamma = \pi$

### Law of sines

$\frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}$ This equivalence may be derived be noting that \begin{align*} \sin \alpha = \frac{h_{c}}{b} &\implies b \sin \alpha = h_{c} \\\\ \sin \beta = \frac{h_{c}}{a} &\implies a \sin \beta = h_{c} \\\\ \end{align*} Giving $a \sin \beta = b \sin \alpha$ Or alternatively $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta}$ With a similar equivalence reachable for $$\frac{c}{\sin \gamma}$$.

### Cosine rule

$a^{2}=b^{2}+c^{2} - 2bc \cdot \cos{\alpha}$

### Pythagorean identities

$\sin^{2}\theta + \cos^{2}\theta = 1$

$\tan^{2}\theta+1 = \sec^{2}\theta$

$\cot^{2}\theta+1=\csc^{2}\theta$

### Other properties

$\cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta$

$\cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta$

$\sin(\alpha + \beta) = \sin \alpha \cdot \cos \beta - \cos \alpha \cdot \sin \beta$

$\sin(\alpha - \beta) = \sin \alpha \cdot \cos \beta + \cos \alpha \cdot \sin \beta$