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 π6\frac{\pi}{6} π4\frac{\pi}{4} π3\frac{\pi}{3} π2\frac{\pi}{2}
sin(x)\sin(x) 0 12\frac{1}{2} 22\frac{\sqrt{2}}{2} 32\frac{\sqrt{3}}{2} 11
cos(x)\cos(x) 1 32\frac{\sqrt{3}}{2} 22\frac{\sqrt{2}}{2} 12\frac{1}{2} 0
tan(x)\tan(x) 0 13\frac{1}{\sqrt{3}} 1 3\sqrt{3} undefined

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

sin(π6)=12\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)\sin(x), as the number being rooted just goes up by one (from 12\frac{\sqrt{1}}{2} to 22\frac{\sqrt{2}}{2} to 32\frac{\sqrt{3}}{2}). The values of cos(x)\cos(x) do the same thing, but the other way round. For tan(x)\tan(x), as

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

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