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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

\[ \cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta \]

\[ \cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \]

\[ \sin(\alpha + \beta) = \sin \alpha \cdot \cos \beta - \cos \alpha \cdot \sin \beta \]

\[ \sin(\alpha - \beta) = \sin \alpha \cdot \cos \beta + \cos \alpha \cdot \sin \beta \]