# 7.2 Spangles

Spangles are "special" angles. They’re special because they show up a lot. Their values are given in this table 44 4 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

$\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$ (7.4)

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

$\tan(x)=\frac{\sin(x)}{\cos(x)}$ (7.5)

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