Question #3c0ea

1 Answer
Jan 3, 2018

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°#.