Question

Question #3c0ea

Answer 1

I assume you mean `\sin 20°`. There is no "simple" value, really; all you can do is punch it into a calculator and get a decimal approximation.

Explanation:

There is actually a geometric aspect to this problem. Certain angles can be constructed with just an unmarked straightedge and compasses (https://en.wikipedia.org/wiki/Compass-and-straightedge_construction), and when we say things like

`\sin 30°=1/2`

`\sin 18°={\sqrt{5}-1}/4`

we can express these results in terms of the geometric constructions. Unfortunately, among angles with whole numbers of degrees only multiples of `3°` have a compass and straightedge construction, and `20°` doesn't work.

In https://en.wikipedia.org/wiki/Trigonometric_functions some values are given for various multiples of `3°`. Note that some expressions are more complicated than others, a fact that correlates with more complicated geometric constructions. For instance, to get an angle of `3°` you basically have to construct `18°` and `15°` angles and then take the difference. That complexity is reflected in the complicated value given for `\sin 3°`.