When does a star rise with the sun? Problem: Suppose an observer's latitude is 45 S and a star's RA/DEC is
3h15min/41S. On what date will the sun set with the star.

I have been stuck on this problem for a while.
I found out that the time of setting of a star is found using the formula:

$$\alpha+\arccos(-\tan(\delta).\tan(\phi))$$
$\implies3\frac{1}{4}^h+\arccos(-\tan(-41)\tan(-45)) = \alpha_{sun}+\arccos(-\tan(\delta_{sun})\tan(-45))$
$\implies 47.8 = \alpha_{sun}+arccos(tan(\delta_{sun}))$

I am not able to solve this any further.
Is there something wrong in my approach? Something I am missing?
If not how do I solve the above equation further?
Thanks in anticipation!
The formula you gave is to find the hour angle of the star while setting, not the setting time itself. Suppose the hour angle of star is $HA\star$, and $RA=\alpha$,then the Local Sidereel Time is given by $LST=HA\star+\alpha$. At the time of setting, $HA\star=10^h1^m30.2^s$. Thus, $LST=13^h16^m30.2^s$at the time of setting.

Now, $RA\odot=6^h$ on $\text{June} 21^{st}$ and $HA_\odot =\arccos(\tan(23.43^{\circ}))=4^h17^m17^s$ at sunset. Thus the star sets with the sun after $\text{ June} 21^{st}$. Since $RA_\odot$ increases approximately by $2^h$ every month, we can assume that the star sets with the sen near $\text{July }21^{st}$. Let it set after $n$ days. At $\text{July }21^{st}$, $\delta_\odot = 20.09^{\circ}$ which gives us $HA_\odot=4^h34^m11^s$, and $RA_{\odot}=8^h9^m44^s$. The $HA_\odot$ at sunset is approximately the same for nearby dates. Since $\delta_\odot$ is also near the maximum, we can approximate $\Delta RA_\odot= \omega_\odot n$. Thus,

$$13^h16^m30.2^s=8^h9^m44^s + \omega_\odot n +  4^h34^m11^s$$
$$ \omega_\odot n =0.5433^h$$
$$n=8$$

At  $\text{July } 29^{th}$, $\delta_\odot=18.23^{\circ}$, $HA_\odot= 4^h43^m6^s$, $RA_\odot=8^h42^m12^s$ which gives us $LST=13^h25^m18^s$, which is a bit more than expected. Trial and error for few days before gives us that on $\text{July } 26^{th}$, the sun sets at LST $13^h15^m9^s$ which would be the most appropriate answer.

Thus, the sun sets along with the star at $\text{July } 26^{th}$