What is the meaning of the title Trinity Seven? I understand why Trinity Seven because there are seven deadly sins represented by a girl which is needed to be controlled by Arata. But I don't understand why Trinity? Which is referred to as the Trinity? Does Arata's magic have three forms? Does he need to do three things to control these seven sins? Are there three magus who can control these seven sins? What does the Trinity represent?
This is what I currently know about the term Trinity Seven and Trinity in the context of this manga.

The accuracy of this post depends largely on the fan translation of the manga series.

When writing this answer, I confirmed all the facts by checking back on the manga. However, I didn't check out the whole series from start to end. I only picked out the chapters which I remembered to have the relevant content and cited from it.

Trinity Seven

Trinity Seven refers to the 7 strongest mage in a world (there can be many parallel worlds).

In Arata's world, the Trinity Seven are the 7 girls: Asami Lilith, Kannazuki Arin, Kazama Levi, Yamana Mira, Fudou Akio, Kurata Yui and Lieselotte Sherlock.

According to Liese in chapter 17, to be a member of the Trinity Seven means that


   one has acquired the Lost Technica in one's archive. In Liese's case, it was "Last Crest".


As explained in chapter 36 of the manga, the magic king is closely related to the Trinity Seven. Below is the summarized definition, based on  the fan translation by VISCANS:


   There exists an infinite number of worlds, with each world connecting to each other by causality. Worlds begin and end, are born and destroyed over and over again. Magic king exists for such system, and his mission, his fate is to reduce his world to nothingness. A magic king is born human, a baby with the secret of unlimited magic power. Regardless of the circumstances of the magic king, he will encounter the Trinity Seven and begin his transformation into a proper magic king.


Trinity

In chapter 36 of the manga,


   ... after Arata copied Lieselotte's Logos Arts in Acedia archive, along with Lilith's Outer Alchemy in Luxuria archive and Akio's Matra Enchantment in Gula archive, he was confirmed by the sword Judecca to have attained Trinity.


However, the condition for Trinity is quite ambiguous. At the beginning of chapter 35,


   ... Arata found the Black Imperial Sword Judecca. The sword identified Arata's Magic King element and confirmed his Superbia archive and Impero thema, then it went on to recognize Outer Alchemy in Luxuria archive and Matra Enchantment in Gula archive. However, the sword concluded that Arata did not meet the Trinity requirement, and it started to initiate Magic King Berserk Mode.


It was quite clear from chapter 5 of the manga that there are many thema (research topic) in an archive. From my personal impression, it seems that different mages working on the same thema may study and master different magics.


   And along that line of reasoning, I speculate that Arata had yet to come up with or mastered any magic under Imperium thema, so it was not counted towards the Trinity requirement.


Apart from the likely direction of Arata eventually copying the magic from all members of Trinity Seven, and thus in control of magics from all 7 archives named after the 7 deadly sins, it is not clear how the Trinity requirement works. We know "trinity", from Christian doctrine, has tri- meaning three in it. However, it is not clear three of what (archive alone is incorrect, as seen above), and whether it requires the mage to have certain level of mastery.

To make the matter more confusing, according to chapter 37 of the manga,


   Arata attained power of a Trinity when Akio, Liese and Arata himself connected to their archives.


And at the end of the chapter,


   Abyss Trinity, Lilith's father and antagonist in the arc, was surprised by the fact that Arata exceeded Trinity after Lilith reconnected to her archive, which means Arata had 4 archives executed at the same time.


So there seems to be a further distinction between achieving the Trinity status and actually performing magic in Trinity mode.