# exact trigonometry tables

## 0.1 Basic angles

Since the trigonometric ratios for most angles cannot be calculated exactly in closed algebraic form, a few well-known angles that can be calculated often comprise the bulk of textbook exercises involving trigonometry.

The basic angles are given in Table 1.

## 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):

 $\displaystyle\sin 15^{\circ}$ $\displaystyle=\sin(45^{\circ}-30^{\circ})$ $\displaystyle=\sin 45^{\circ}\cos 30^{\circ}-\cos 45^{\circ}\sin 30^{\circ}$ $\displaystyle=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}% \cdot\frac{1}{2}$ $\displaystyle=\frac{\sqrt{6}-\sqrt{2}}{4}\,.$

Likewise, we can find that

 $\displaystyle\cos(15^{\circ})$ $\displaystyle=\frac{\sqrt{6}+\sqrt{2}}{4}$ $\displaystyle\sin(75^{\circ})$ $\displaystyle=\sin(45^{\circ}+30^{\circ})=\frac{\sqrt{6}+\sqrt{2}}{4}$ $\displaystyle\cos(75^{\circ})$ $\displaystyle=\frac{\sqrt{6}-\sqrt{2}}{4}\,.$

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

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

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 obtained by 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

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

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.

 Title exact trigonometry tables Canonical name ExactTrigonometryTables Date of creation 2013-03-22 15:31:26 Last modified on 2013-03-22 15:31:26 Owner stevecheng (10074) Last modified by stevecheng (10074) Numerical id 11 Author stevecheng (10074) Entry type Example Classification msc 51M04 Classification msc 33B10 Related topic ConstructibleAnglesWithIntegerValuesInDegrees Related topic Cotangent2 Related topic Trigonometry Related topic TheoremOnConstructibleAngles Related topic GraphOfEquationXyConstant