I'm familiar with the idea of using algebra to study certain types of plane curves, and my understanding is that there is a whole class of "algebraic curves" that can be studied this way. It would also make sense that there are other plane curves, such as $y - \sin(x)=0$, that are not algebraic curves. Are there nevertheless algebra-like ways of studying these curves, and if not, why?
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Very much depends on the level. Many basic properties of curves are studied using the differential and integral calculus. If you are using techniques from the calculus, it does not matter very much whether the curve is algebraic or transcendental.