Curves & Motion · Superchallenge problem

A plane curve starts at the origin with a horizontal tangent, heading in the positive $x$-direction, and bends to the left (upward). At every point its curvature equals $\cos\varphi$, where $\varphi$ is the angle from the positive $x$-axis to the tangent direction. Find the $y$-coordinate of the point of the curve with $x = \dfrac{\pi}{3}$.