Biot-Savart magnetic field at point $P$ semi-infinite wire

I have an infinite wire and I would like to know the magnetic field at a point $P$.

$$vec{dB} = frac{u_0I}{4pi R^2} cdot vec{dl} times hat{r}$$

for a semi-infinite wire as in the image I know $theta in[0,90]$, but I’m not sure how to write it mathematically. I know at the end the equation should looks like

$$vec{B} = frac{u_0I}{4pi R}(sin(90^circ)+sin(0^circ))$$

However, I’m not sure how to get it from this properly.
$$vec{B} = frac{u_0I}{4pi R^2} cdot intvec{dl} times hat{r}$$