Since the sides are sin(x), cos(x), and tan(x), and
the triangle is isosceles, two of the sides are equal, 
but we don't know which two.  So we try all three 
possibilities:

    1.  We try sin(x) = cos(x) 
Divide both sides by cos(x)
                = 1
                    x = 45° or 

so the sides would be 

               sin() = 
               cos() = 
               tan() = 1

But that is a right triangle, and don't want that answer.

    2.  We try sin(x) = tan(x) 
That can only have solution x=0,
But those cannot be angles of a triangle.

    3.  We try cos(x) = tan(x)
               cos(x) = 
Multiply through by cos(x)
              cos²(x) = sin²(x)
          1 - sin²(x) = sin²(x)
-sin²(x) - sin(x) + 1 = 0
Multiply through by -1
 sin²(x) + sin(x) - 1 = 0   
  
               sin(x) =  
               sin(x) = 
               sin(x) = 
               sin(x) = 
sin(x) must be positive, so           
               sin(x) =  (a.k.a "the golden ratio")

Drawn to scale here is that triangle:



Figures involving the golden ratio are supposed to be "the most
beautiful".  So maybe this triangle is "the most beautiful one."  

:-)

Edwin