In what context are these mathematical symbols used?

$\oplus$ and $\otimes$ are used to denote direct sum and tensor product, which are ubiquitous in mathematics. They also have a pedagogical use as symbols for addition in an abstract abelian group resp. multiplication in an abstract group when you want to make the point that groups are much more general than addition and multiplication of real or complex numbers. $\ominus$ is occasionally used in this abstract way as well.

Your "sine wave symbol" looks to me very similar to $\sim$. Certainly that is used: $$ X\sim N(\mu,\sigma^2) $$ $X$ is a normally distributed random variable with expectation $\mu$ and variance $\sigma^2$. $$ X\sim \operatorname{Bin}(n,p) $$ $X$ is a binomially distributed random variable with parameters $n\in\mathbb{N}$ and $p\in[0,1]$.

$$ f(x) \sim \sum_{n=-\infty}^\infty c_n e^{inx} $$ The Fourier series of $f(x)$ is that series. There is no commitment to saying the series converges to $f(x)$ (in some cases it does; in others it doesn't, and it can depend on which kind of convergence is being considered).

$$ a \sim b $$ $a$ is related to $b$ (just which sort of "relation" is referred to depends on the context).

If $A \subset B$ are linear subspaces of a Hilbert space, $B \ominus A = \{x \in B: (x,y) = 0 \text{ for all }y \in A\}$. $\ominus$ is also used for the symmetric difference of sets.