I see that a vector can be described in spherical co-ordinates, with respect to it's cartesian co-ordinates as
$$ \mathbf{r}=r\sin{\theta}\cos{\theta}\mathbf{\hat{x}}+r\sin{\theta}\sin{\theta}\mathbf{\hat{y}} + r\cos{\theta}\mathbf{\hat{z}}$$
So, the unit vector $\mathbf{\hat{r}}$ is calculated by dividing the vector $\mathbf{r}$ by it's magnitude $r$.
$$ \mathbf{\hat{r}}=\sin{\theta}\cos{\theta}\mathbf{\hat{x}}+\sin{\theta}\sin{\theta}\mathbf{\hat{y}} + \cos{\theta}\mathbf{\hat{z}}$$
But I am unsure as to how to get the other two unit vectors, $\mathbf{\hat{\theta}}$ and $\mathbf{\hat{\phi}}$.
Using the reasoning above I thought I could just add 90 degrees to either $\theta$ or $\phi$ to the $\mathbf{\hat{r}}$ result to get the corresponding unit vector. This reasoning gives me a correct result for $\mathbf{\hat{\theta}}$. However, for the result I have been given, $\mathbf{\hat{\phi}}$ has only $\mathbf{\hat{x}}$ and $\mathbf{\hat{y}}$ components. Why is this?