The projection of the particle's coordinate onto the $x$ axis is given by $$x = R\cos\theta\nonumber$$ and the projection onto the $y$ axis by $$y = R\sin\theta\nonumber$$ We also know from mechanics that the velocity $v$, which is constant, is the tangential velociy, and is related to the angular velocity $\omega$ by $v=R\omega$. Since $R$ is also constant, then that means the angular frequency $\omega$ is constant, and $\omega$ is the rate of change of the angle $\theta$, which means $$\omega = \frac{d\theta}{dt}\nonumber$$ is constant. This is a simple differential equation, solved by $\theta = \omega t$, which means that our equations for $x$ and $y$ become $$\begin{align} x &= R\cos\omega t\nonumber\\ y &= R\sin\omega t\nonumber \end{align}\nonumber$$