Improved Spanning on Theta-5

We show an upper bound of $\frac{ \sin\left(\frac{3\pi}{10}\right) }{ \sin\left(\frac{2\pi}{5}\right)-\sin\left(\frac{3\pi}{10}\right) } <5.70$ on the spanning ratio of $\Theta_5$-graphs, improving on the previous best known upper bound of $9.96$ [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]