You can get that the interior line is shorter and the outer line is longer than the arc by relating these to the inner and outer polygons used in Archimedes' de

You can get that the interior line is shorter and the outer line is longer than the arc by relating these to the inner and outer polygons used in Archimedes' demonstration that the area of a circle may be related to the circumference: http://www.ams.org/publicoutreach/feature-column/fc-2012-02 http://www.ams.org/publicoutreach/feature-column/fc-2012-02 It is possible to prove that the circumference of the outer polygons gives a decreasing sequence and the circumference of the interior polygons gives an increasing sequence simply by applying the triangle inequality when you double the number of sides. This is sufficient to justify the use of the squeeze theorem here, although you are correct that the axiomatization of geometry is quite subtle. I did not give much thought to the definition of the trigonometric functions while writing this because I assumed this theory is usually fully developed before we begin complex analysis (but is it?).