Let $\mathcal{L} = \{f,g\}$ where $f$ is binary and $g$ unary. Consider the $\mathcal{L}$-structure $\mathfrak{M}$ with underlying set $\mathbb{R}$ and $f^{\mathfrak{M}}$ is the default multiplication, $g^{\mathfrak{M}}$ is the sine function.
Give an $\mathcal{L}$-formula $\phi(x)$ with one free variable $x$ such that $\forall a\in \mathbb{R}$: $$\mathfrak{M} \vDash \phi(a) \Leftrightarrow \text{there is an $n\in \mathbb{N}$ such that $a=(2n+0.5)\pi$}$$
I guess the missing constant symbols create a problem here? I was thinking of using something like to avoid the $1$-symbol, but now I'm using the $0$-symbol... How can I solve this problem?
$$\phi(x) = \exists y(y\not = 0 \wedge f(y,y)=y \wedge g(x) = y)$$