7.2 Spangles ‣ Chapter 7 Trigonometry ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 7 Trigonometry ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

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 $\sin(x)$ $\cos(x)$ .