Re: sides of a right triangle with 15-deg (and 75 deg) is somewhat well known

On Tue, 27 May 2025 18:15:56 +0000, HenHanna wrote:

I only learned a few days ago....   that the ratio of  the sides of a
right triangle with 15 degree angle   (and 75 deg)   is somewhat well
known.
 
 
         (I only knew about the triangle with the sides  1, 2, and Sqrt3
)

I suspect there'll be an exact value for the ratio of the triangle's
sides, based on the geometry of the appropriate regular polygon
(equilateral triangle, square, pentagram, hexagram etc.). The solution
will appear more significant when the included angles happen to be an
integer number of degrees.

So, for an equilateral triangle there is an exact solution for the side-
ratio for a triangle of internal angles 30, 60, 90. I think you could then
use the half-angle formula to derive the exact side-ratio for a 15, 75, 90
triangle.

The square provides an exact side-ratio for a 45, 45, 90 triangle, which
could be extended, but isn't very appealing as 27.5 isn't an integer
number of degrees.

The pentagram provides an exact solution for a 36, 54, 90 triangle, which
can be extended to a 18, 72, 90 triangle.

You may be able to continue with regular polygons of greater number of
sides, but the solution will generally become more complicated and the
results less appealing.

If anyone knows the above to be incorrect, then feel free to correct the
error. I am speculating.

--
David Entwistle