exact trigonometry tables

0.1 Basic angles

Since the trigonometric ratios for most angles cannot becalculated exactly in closed algebraic form,a few well-known anglesthat can be calculatedoften comprise the bulk of textbookexercises involving trigonometry.

The basic angles are given in Table 1.

Table 1: Basic angles encountered in trigonometry

0.2 Other angles by addition and halving

These basic angles can be easily extended to obtain more angles of interest. Adding multiples of ${90}^{\circ }$ merely rotates these angles into other quadrants; the appropriate values of $\sin$ and $\cos$ can be obtained through symmetry.

The values for ${15}^{\circ }$ can be obtained by using the formula for the difference of angles (http://planetmath.org/AngleSumIdentity):

$\sin {15}^{\circ }$$=\sin ({45}^{\circ }-{30}^{\circ })$

$=\sin {45}^{\circ }\cos {30}^{\circ }-\cos {45}^{\circ }\sin {30}^{\circ }$

$={\displaystyle \frac{\sqrt{2}}{2}}\cdot {\displaystyle \frac{\sqrt{3}}{2}}-{\displaystyle \frac{\sqrt{2}}{2}}\cdot {\displaystyle \frac{1}{2}}$

$={\displaystyle \frac{\sqrt{6}-\sqrt{2}}{4}}.$

Likewise, we can find that

$\cos ({15}^{\circ })$$={\displaystyle \frac{\sqrt{6}+\sqrt{2}}{4}}$

$\sin ({75}^{\circ })$$=\sin ({45}^{\circ }+{30}^{\circ })={\displaystyle \frac{\sqrt{6}+\sqrt{2}}{4}}$

$\cos ({75}^{\circ })$$={\displaystyle \frac{\sqrt{6}-\sqrt{2}}{4}}.$

More exact angles can be obtained by solving the double angle identity:

$$\sin \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{2}},\cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}.$$

So for example, $\sin {7.5}^{\circ }=\sqrt{(4-\sqrt{6}-\sqrt{2})/8}$. These angles can be further added and subdivided to obtain a dense subset of exactly known angles. However, such effort is not generally useful. Computers and calculators use a combination of lookup-tables and numeric iteration to obtain their values.

0.3 The angles ${18}^{\circ }$, ${36}^{\circ }$, ${54}^{\circ }$, ${72}^{\circ }$

The ${18}^{\circ }$-${36}^{\circ }$-${54}^{\circ }$-${72}^{\circ }$ series of angles cannot be obtainedby halving, doubling, adding or subtracting the previous angles. Nevertheless, they are constructible, and their exact values can be derived by the following elementary procedure:

Consider an isosceles triangle with the angles ${72}^{\circ }$, ${54}^{\circ }$ and ${54}^{\circ }$.From the triangle we derive the relation:

$$\sin \frac{{72}^{\circ }}{2}=\cos {54}^{\circ }$$

Notice that $72=4\times 18$ and $54=3\times 18$, so if $x={18}^{\circ }$,then

$\sin 2x$$=\cos 3x$

$2\sin x\cos x$$=4{\cos }^{3}x-3\cos x$

$2\sin x$$=4{\cos }^{2}x-3$

$2\sin x$$=4(1-{\sin }^{2}x)-3$

The last equation is a quadratic equation that can be solved for $\sin {18}^{\circ }$. Carrying out the calculations, we obtain the values in Table 2.

Table 2: Other constructible angles in trigonometry