alt Hacker News> This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known.
> Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.
But that's not how sine was introduced. It's been around since classical geometry. It was always easy to solve the differential equation y'' = -y, because the sine had that property, and we knew that.
Heck, you can tell this just by looking at the names of the functions you mentioned. "Sine" is called "sine", which appears to have originated as an attempted calque of a Sanskrit term (referring to the same function) meaning "bowstring".
"Square root" is named after the squaring function that was used to define it.
Introducing an answer-by-definition gives us negative numbers, rational numbers, imaginary numbers, and nth roots... but not sines, come on. You can just measure sines.
You can calculate, measure, draw, construct, write a power series for, express as hypergeometric function, etc. the Bring radical too.
All of these concepts, from sine to real numbers, Bring radicals to complex exponentials, can all be defined in different, equivalent ways. What is interesting are the properties invariant to these definitions.
It still doesn't seem to me that a square root should be any more or less contrived than a Bring radical. Maybe we should call it a ultraradical instead?