It depends on your context, and is mentioned in the article. The advantage of turns comes from the fact that the implementation of sinᵣ etc. is internally doing a conversion to turns, so by using sin_turns directly, you avoid calculating π/π with every call.

It’s not a call for someone doing calculus or solving differential equations to abandon radians, just for the particular case of getting a numerical value for sin, cos, etc. from a library, having direct access to a turns-based function would produce faster code and also avoid some of the rounding errors that come from that π/π not to mention the imprecision of any angle that isn’t 0.