Wedges in a triangle 2

If there's a closed form I don't plan find it. See the complicated function to be maximized below. The maximum occurs when the angle is about $79.73791977^{\circ}$

Let the sectors have unit radius then the total area of the sectors is $\large A(\theta)=3\cdot \theta \cdot \frac{\pi}{360^{\circ}}$ where $\theta$ is the angle in degrees.

The formula for the side length is $\large S(\theta)=\frac{2\sqrt{3}\sin{\theta}-2\cos{\theta}+\sqrt{3}}{2\sin{\theta}}$

The proportion to be maximized is then $\LARGE P(\theta)=\frac{A}{\frac{\sqrt{3}}{4}S^{2}}$

Which I first maximized with Geometer's Sketchpad and then confirmed with my graphing calculator which gives more digits of accuracy.