Solution for $n=2$: In view of the rotational invariance, the distribution the distance of the random point $(C,S)$ from the origin is the same as that for $(1,0)+(\cos U,\sin U)=(1+\cos U,\sin U)$, where $U$ is uniformly distributed on $[0,2\pi]$. So, $R$ is equal in distribution to $\sqrt{(1+\cos U)^2+\sin^2 U}=2|\cos U/2|$ and hence to $2\cos V$, where $V$ is uniformly distributed on $[0,\pi/2]$. So, the pdf $f_R$ of $R$ is given by $$f_R(r)=\frac2{\pi\sqrt{4-r^2}}1_{0<r<2} $$ for real $r$.
Consider also the case of large $n$. Note that $R=R_n$ is the length of the vector $S_n$ that is the sum of $n$ iid copies of the random vector $X:=(\cos U,\sin U)$, with $U$ as above. The mean of $X$ is $(0,0)$ and its covariance matrix is $\frac12\,I_2$, where $I_2$ is the $2\times2$ identity matrix. So, by the multivariate central limit theorem, the distribution of $\sqrt{\frac 2n}S_n$ converges to the standard bivariate normal distribution. So, the distribution of $\sqrt{\frac1n}R_n$ converges to the Maxwell distribution, with pdf $f_M$ given by $$f_M(r)=2r e^{-r^2}1_{r>0} $$ for real $r$.