Definition of image of an element; what does it mean to say that a function takes all values of subset?

Question

Wikipedia Article on topic of images that

" Given $y$, the function $f$ is said to "take the value $y$ " or "take $y$ as a value" if there exists some $x$ in the function's domain such that $f(x)=y$. Similarly, given a set $S, f$ is said to "take a value in $S^{\text {" }}$ if there exists some $x$ in the function's domain such that $f(x) \in S$. However, " $f$ takes [all] values in $S^{\text {" }}$ and " $f$ is valued in $S^{\prime \prime}$ means that $f(x) \in S$ for every point $x$ in $f^{\prime}$ s domain."

My Query
It is related to the the meaning of $"f$ take all values in $S"$. Does this means $S$ is the range? Or $S$ is a subset of actual range of $f?$ In my opinion, former one is correct because it is being said that for all $x$ belongs to $f$, domain $f(x)$ is belonging to $S$ or is it that $S\subseteq f(x)$ range?

Answer 1

$S$ is the range of $f(x)$. For example, for $sin(\theta$) , $S=\mathbb C$ (yes, you can define $sin(\theta)$ for the complex numbers).