If you know the lengths of the two legs of a right triangle and you are trying to solve for the measures of the acute angles of the triangle without calculating the length of the hypotenuse, the most useful trigonometric ratio would be the tangent. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. In this case, you could take the ratio of the shorter leg to the longer leg to find the tangent of one of the acute angles. You could then use the inverse tangent function to find the measure of that angle.
Alternatively, you could use the reciprocal trigonometric ratio of the cosine. The cosine of an acute angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. Since we do not know the length of the hypotenuse, we cannot use the cosine directly. However, the reciprocal of the cosine is the secant, which is the ratio of the hypotenuse to the adjacent side. By taking the reciprocal of the ratio of the longer leg to the shorter leg, we can find the secant of one of the acute angles. We could then use the inverse secant function to find the measure of that angle.