The definition of the Fourier transform for three dimensions is
$$\mathcal{F}[f(\mathbf{r})](\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{r}}f(\mathbf{r})\,d^3 r$$
If the function $f(\mathbf{r})$ is spherically symmetric (i.e. f($\mathbf{r}$) can be written as $f(r)$, where $r=|\mathbf{r}|$), the transform can be written as
$$\mathcal{F}[f(\mathbf{r})](\mathbf{k})=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}e^{-i|\mathbf{k}|r\cos (\theta)}f(\mathbf{r}) r^2\sin(\theta) dr d\theta d\phi$$
This is what is given in my book, but I have no idea where this $\cos(\theta)$ term came from. I read this math.SE question and answer, but I still don't get why the $\cos(\theta)$ term is there.
In the triple integral, $\theta$ is designated with respect to the $z$-axis and the $\mathbf{r}$ vector. I don't see why it is also claimed that it is the angle between the $\mathbf{r}$ vector and the arbitrary $\mathbf{k}$ vector.
Could you help clear up my confusion?