# A calculus of the absurd

#### 8.3 Spangles

Spangles are "special" angles. They’re special because they show up a lot. Their values are given in this table 5959 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

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

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

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

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