%path = "maths/trigonometry" %kind = kinda["texts"] %level = 11

In the following drawing we have a circle with radius 1. The length of the arc on such a circle is a measure for the angle. It is called radian and the unit is rad. The right angle (90°) is \(\pi/2\). In general

\[\frac{\pi}{180}\alpha[°] = \alpha[rad]\]

By changing this angle one also changes the lengths labeled with \(\sin\alpha\), \(\cos\alpha\) and \(\tan\alpha\). These lengths are determined by the angle, which is equivalent to say that the lengths are functions of the angle.

\coordinate (O) at (0,0);
\coordinate (C) at ({2*cos(60)},{2*sin(60)});
\coordinate (P) at ({2*cos(60)},0);
\coordinate (D) at (2,{2*tan(60)});
\draw[black, very thin] (O) circle [radius=2];
\draw[red,thick] (2,0) arc [radius=2, start angle=0, end angle=60] node[midway,above]{\tiny $\alpha$};
\draw[blue,thick] (O) -- (C) node[midway,above]{\tiny $1$};
\draw[blue,thick] (P) -- (C) node[midway,right]{\tiny $\sin\alpha$};
\draw[blue,thick] (O) -- (P) node[midway,below]{\tiny $\cos\alpha$};
\draw[green,thick] (P) -- (2,0);
\draw[green,thick] (2,0) -- (D) node[midway,right]{\tiny $\tan\alpha$};
\draw[green,thick] (C) -- (D);
\draw[xshift=-1.1cm,yshift=-1cm] node[right,text width=2.2cm]
{ \tiny $\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$\\$\sin^2\alpha+\cos^2\alpha=1$ };

Similarity = affine mapping

One can scale this diagram to an actual rectangular triangle and completely determine it by knowing one sharp angle and one side.

The graphs of the above trigonometric functions are as follows

\begin{axis}
    [
    ymin=-1,ymax=1,
    xmin=0,xmax=2*pi,
    xtick=\empty,
    ytick={-1,0,1},
    extra x ticks={1.5708,3.1416,4.712,6.2832},
    extra x tick labels={$\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$},
    every extra x tick/.style={
            xticklabel style={anchor=north west},
            grid=major,
            major grid style={thick,dashed,red}
    },
    axis lines = center,
    xlabel=$x$,ylabel=$y$,
    enlargelimits=0.2,
    domain=0:2*pi,
    samples=100,
    axis equal,
    ]
    \addplot [green,thick] {tan(deg(x))} node [midway,left]{tan};
    \addplot [red,thick] {sin(deg(x))} node [above]{sin};
    \addplot [blue,thick] {cos(deg(x))} node [above]{cos};
\end{axis}

Some values of the functions can be found via calculations on the equal sided triangle (\(\pi/3\), \(\pi/6\)) or on a square with side length 1 (\(\pi/4\)).

\(\cos\) is symmetric: \(\cos(-\alpha)=\cos\alpha\)

\(\sin\) is antisymmetric: \(\sin(-\alpha)=-\sin\alpha\).

All trigonometric functions have a period of \(2\pi\): \(sin|cos|tan(\alpha+2\pi)=sin|cos|tan(\alpha)\).

Because the sharp angles of the rectangular triangle add to \(\pi/2\), we have

\[\begin{split}\sin(\pi/2 - \alpha)=\cos\alpha\\ \cos(\pi/2 - \alpha)=\sin\alpha\end{split}\]