To find the magnitude and direction of the acclleration for the uniform circular motion,we consider the below figure
Where in particle $p$ moves at constant speed $v$ around a circle of radius $r$.At the instant shown,$p$ has coordinates $x_{p}$ and $y_{p}$. since velocity $\vec{v}$ of a moving particle is always tangent to the particle's path at the particle's position.In the above figure that means $\vec{v}$ is perpendicular to a radius $r$ drawn to the particle's position.
How does the angle $\theta$ that $\vec{v}$ makes with a verticle at $p$ equals the angle $\theta$ that radius $r$ makes with the $x$-axis?