Recently I am investigating some geometric construction (i.e., compass and ruler constructions) and encounter into the issue of finding sine for some angles. I remember most whole number degree angles could be expressed in surd forms, only it is less pretty than the standard angles of 30°, 45°, 60°. Mostly I would start from the standard angles and break down to the degree I am interested by half angle formula or angle sum and difference formula. But some angle could result in surd under surd, for example:

\[\sqrt{2 - \sqrt{3}}\]

Whether we can simplify this or express it in another format is the issue.

Let us start with the example. First consider if we want to factor it out into a product. Obviously it is not possible to be a product of two surd forms, so let us assume

\[\sqrt{2 - \sqrt{3}} = \sqrt{a}(\sqrt{b}-c).\]

We use \(\sqrt{b}-c\) instead of \(\sqrt{b}+c\) because we observed the negative sign in \(-\sqrt{3}\). We may also try with \(\sqrt{2 - \sqrt{3}} = \sqrt{a}(c-\sqrt{b})\) but we will know in the following if this is actually the case. Move the bracket term to under the surd gives:

\[\begin{aligned} \sqrt{2 - \sqrt{3}} &= \sqrt{a}(\sqrt{b}-c) \\ &= \sqrt{a (b-2c\sqrt{b}+c^2)} \\ &= \sqrt{ab+ac^2-2ac\sqrt{b}} \end{aligned}\]


\[\begin{aligned} ab+ac^2 &=2\\ 2ac &= 1\\ b &= 3. \end{aligned}\]

Expressing \(c\) in terms of \(a\) gives the quadratic equation \(12a^2-8a+1=0\) and solving this gives \((a,b,c)=(\frac{1}{2},3,1)\) or \((a,b,c)=(\frac{1}{6},3,3)\). These two, indeed, corresponds to

\[\sqrt{2 - \sqrt{3}} = \sqrt{\frac{1}{2}}(\sqrt{3}-1) = \sqrt{\frac{1}{6}}(3-\sqrt{3})\]

which are the same (and explains why we assume the form \(\sqrt{a}(\sqrt{b}-c)\) is enough).

Another approach1 is not to factor it out but express into sum of two surds:

\[\sqrt{2 - \sqrt{3}} = \sqrt{a} - \sqrt{b}.\]

Again, we take \(\sqrt{a}-\sqrt{b}\) instead of \(\sqrt{a}+\sqrt{b}\) because we observed the negative sign of \(-\sqrt{3}\). This time, we squared both side and found that \(2 - \sqrt{3} = a + b - 2\sqrt{ab}\). By solving \(a+b=2\) and \(4ab=3\), we have \((a,b)=(\frac{1}{2},\frac{3}{2})\) or \((a,b)=(\frac{3}{2},\frac{1}{2})\). Because we take only positive values for the surd, comparing the expression for correct polarity we found that the solution should be

\[\sqrt{2 - \sqrt{3}} = \sqrt{\frac{3}{2}} - \sqrt{\frac{1}{2}}.\]

It turns out, simplifying surds directly does not seems possible but we can easily do it if we first assume the form we want to obtain. Here contains a list of sine values for all integer angles from 0° to 90°, but the surd forms are ugly. Probably we can simplify them using this approach.

  1. This form is described in Tutorial Algebra Volume 1