Yes but it is obfuscating the fact that sine tables or calculator just make it easier. Who the heck cares what the cosine value is, what people care about is the dimensions of a,b,c and the angles A,B & C. The actual Law of Cosines is: c^2 = a^2 + b^2 -2ab(cos C), thereby giving the ability to determine the length of c, a or b plus, if anyone cares, the value of cos C.
Right, but in standard geometry the basic unit is the angle, not the cosine of the angle, and there is no way to go numerically between the two without a calculator or tables. This reformulation avoids having any angle variable in any formula, so you can always calculate the answer in terms of the initial data without having to use calculators or tables and without the inevitable inexactness that results.
(You may end up at the end with some square roots, but they are very easy to calculate by hand to many decimal places, unlike sines and cosines.)