Integer-sided triangles with trisectible angles are a special class of triangles. Cosines of angles of such triangles are rational numbers with the additional property of representation by a specific third degree polynomial. It is shown that no equilateral triangles and no right angled isosceles triangles belong to that class. In this paper, the emphasis is placed on scalene integer-sided triangles with trisectible angles, but the existence of right triangles with these properties is thoroughly investigated, too, as well as the inclusion of isosceles triangles without a right angle in that class. It is shown that the key feature for the existence of the considered triangle type is the so-called residual. Residual is a positive integer that is in a simple way associated to the rational value of the cosine. More precisely, two acute angles with rational cosines have the same residual if and only if there is an integer-sided triangle containing these two angles. Furthermore, a general formula based on residuals is derived for the sidelengths of the observed scalene triangles without a right angle. The main result of this paper reveals that for each square-free positive integer r there exist infinitely many distinct scalene integersided triangles with trisectible angles and with r as their common residual. Some simple methods for generating such triangles are presented and examples for the corresponding sidelengths are given. Finally, it is shown that the acute angles of primitive Pythagorean triangles are trisectible if and only if their hypotenuse is a perfect cube.