A calculus of the absurd

7.2 Spangles

Spangles are "special" angles. They’re special because they show up a lot. Their values are given in this table 5252 There are numerous problems with the formatting of this table which I will one day get around to fixing..

0 \(\frac {\pi }{6}\) \(\frac {\pi }{4}\) \(\frac {\pi }{3}\) \(\frac {\pi }{2}\)
\(\sin (x)\) 0 \(\frac {1}{2}\) \(\frac {\sqrt {2}}{2}\) \(\frac {\sqrt {3}}{2}\) \(1\)
\(\cos (x)\) 1 \(\frac {\sqrt {3}}{2}\) \(\frac {\sqrt {2}}{2}\) \(\frac {1}{2}\) 0
\(\tan (x)\) 0 \(\frac {1}{\sqrt {3}}\) 1 \(\sqrt {3}\) undefined

Don’t memorise the table! All you need to remember is that

\begin{equation} \sin \left ( \frac {\pi }{6} \right ) = \frac {1}{2} \end{equation}

From there, you can work out the rest of the values for \(\sin (x)\), as the number being rooted just goes up by one (from \(\frac {\sqrt {1}}{2}\) to \(\frac {\sqrt {2}}{2}\) to \(\frac {\sqrt {3}}{2}\)). The values of \(\cos (x)\) do the same thing, but the other way round. For \(\tan (x)\), as

\begin{equation} \tan (x) = \frac {\sin (x)}{\cos (x)} \end{equation}

the values of \(\tan (x)\) can be computed from the values of \(\sin (x)\) \(\cos (x)\).