I'm going to assume you're representing your world as geometric primitives (points, lines, curves, polygons, etc.) rather than sprites. If you're using sprites and per-pixel collision detection, there's nothing special about odd-shaped objects. See this question for some ideas.
Using geometric primitives, the first step is to use a broad-phase algorithm to narrow down the possible pairs of objects that might intersect. See this Stack Overflow question. Checking for collisions between each candidate pair is called the narrow phase, and you have a couple of options.
The simplest solution is to just approximate your curve using a series of line segments. If the curve is a standalone object, you'd just treat the segments as you would any other line segments in the world. If the curve is the edge of some larger object, then as a pre-processing step you'd split the larger object into smaller convex shapes using a process similar to the trapezoid rule from basic calculus. The general term for this process is "convex decomposition", and computing an optimal decomposition of arbitrary polygons is NP-complete, but a good-enough or hand-crafted decomposition will work just fine. The smaller convex subparts can be easily checked for collisions using any of a number of algorithms.
A more difficult solution is to write a custom collision algorithm for collisions between each kind of geometric primitive you want to support. Assuming you already support collisions between points, line segments, rectangles, circles, and convex polygons, adding a curve like your example would require coding a custom algorithm for the following pairs:
- Point - sine curve
- Line segment - sine curve
- Rectangle - sine curve
- Circle - sine curve
- Convex polygon - sine curve
- Sine curve - sine curve
I wouldn't recommend this approach because it's complex and ends up being a lot of work. However, if you can figure out a way to represent your objects using only convex shapes, then the generalized GJK algorithm will work, even if your objects have curves.