- often you use decimal fraction on output and input anyway
- it's slower, even slower with bigint
- no square root, sin, and so on with rationals
Better statement would be "no standard definition possible for square root, sin, and so on with rationals", as rationals with unbounded denominators don't have an inherent precision limit---something like ulps in floating point systems---and the max denominator wouldn't be a good choice for precision even when denominator is limited. Every such function now has to be given an explicit denominator value or similar strategy, which complicates both the interface and implementation.
Are you thinking about a specific library? You aren't the only person who commented this way. But, the truth is that root, sin and so on don't "work" with floats either. In fact, there are common ways to implement these functions by either using tables (which are approximate) or algebraic approximations (that give you... drum roll: rationals!)
But, really, there isn't any way (except symbolically) to represent transcendental functions in computers. It doesn't matter what kind of number you choose to do it.