Using Pythagorean Triangles to Approximate Angles

Abstract

A Pythagorean triangle is a right triangle with integer sides. A well-known example is the triangle with sides 3, 4, and 5. Unfortunately, the angles of these triangles are not as nice as the sides: whether expressed in degrees or radians, they are always irrational. If a Pythagorean triangle has an angle of A degrees and IA 201 < 1/100, we shall say that the triangle 20 degrees to within one hundredth. Let B be any given angle in degrees with 0 < B < 90. Let e be any real number such that 0 < e < 1, e < B, and e < 90 B. The question we answer in this note is: what is the smallest Pythagorean triangle which approximates B degrees to within