Why my phone battery has four terminal? what is the use of the fourth?
Connector A seems a U.FL-R-SMT-10 and probably is for an external/additional antenna.


  U.FL, is a miniature coaxial RF connector for high-frequency signals
  up to 6 GHz manufactured by Hirose Electric Group in Japan.1 U.FL
  connectors are commonly used inside laptops and embedded systems to
  connect the Wi-Fi antenna to a Mini PCI card. Female U.FL connectors
  are not designed with reconnection in mind, and they are only rated
  for a few reconnects before replacement is needed. The female U.FL
  connectors are generally not sold separately, but rather as part of a
  pigtail with a high-quality 1.32 mm doubly shielded cable, which
  allows for a low-loss connection.


(from Wikipedia)