It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.

This is the case of sines & cosines of the following angles in degrees :

30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5,

and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only.

This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.

It is a remarkable fact that all angles counted in degrees as powers of 2 →

n

2

such as:

2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.

have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.

The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 →

n

5.2

such as:

5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.

The purpose of this article is double:

1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.

2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg).

Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.

This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.