The Trinity isn't irrational at all. However it is a revealed truth. There simply isn't anything like it in any other system of belief in the world. If you believe that pagan trinities of gods are the same thing than that is simply a misunderstanding on your part. The Trinity teaches us everything about Judeo-Christinaity as the Trinity is a family unto Itself and Judeo-Christianity has to be understood through the successive covenants in which God, i.e. the Trinity, has attempted to bring us into his own family life, as intimates. From my perspective now as a convert to Catholicism, i.e. Christianity, the whole world and every family in the natural world, of both men and animals, is a reflection of this reality. The Trinity also leads to some very beautiful implications such as truth and love being one and the same thing--not only are they the same but they provide for a theory of objective truth/love which is grounded in God's ontological encounter with himself.
One thing to understand, and unfortunately psychology speaks a different language so this reality of our psychological being isn't reflected in secular psychological theories, is that there is a difference between persons and essences. For us, our essence and person are pretty much the same thing so it is not immediately apparent that this needn't be the case. However even we as humans, from a Christian perspective still do share something of a common essence with one another. This is what makes human communion, and therefore community, possible in the first place. It is also how we all came to lose Sanctifying Grace through activity of single person and how we were restored to it through the actions of another single person.
Initially upon my investigation in Catholicism, coming from a Baha'i perspective, I thought the Trinity was sheer irrationality, and I would never be able to accept it. In fact I use to argue, erroneously, it was a later invention of Christianity. Once however I came to actually understand, in the partial way we are limited to understanding it, I saw that it must always be the logical consequence of a God in whose image we are made. I reiterate the simplest formula below:
God the source and ground of all Being exists, non-contingently, in absolute simplicity, as pure act. The ground of everything is self aware, and this self awareness of God, is itself, a person. This is not very different than the narratives of ourselves that constitute the sum total of our own egos. Essentially our egotistic conception of ourselves is a story that defines who we believe we are. In God's case this Story, is a person who is a perfect, infinite fidelity of Himself, and so God comes to a perfect knowledge of His own infinite self nature through this Second Person, aka the only Begotten Son, aka the Word of God. If there is a self aware God then a Son is begotten of logical necessity and has always existed along with the Father who has begotten Him. Additionally, as soon as the Son gazes back upon the Father, encountering His own ontological reality, Truth and Love are born and are themselves personified this is a Third Person, the Holy Ghost, and since all three persons are of the one essence they are one and the same with each other. Consequently whatever the Holy Spirit is the Father and Son must must be as well. Quite frankly once this is truly grasped it becomes apparent that a God, in whose image we are made, who is not Triune is simply not logically feasible.
The three Persons of the Trinity live in total self sacrifice (love) to one another. Jesus who incarnated into the flesh (here He has become a single person with two essences--in fact I will assert that if you deny the reality of the Trinity you would then have to deny that Jesus had two natures. If this was the case than our human nature hasn't really ascended into heaven and it is not at all clear then what it would then mean to be saved. Certainly such denials have always been denounced as heresy by the Church) and is also known and the Eternal Covenant made flesh has taught us the depths of what such self giving means as he gave himself for us as St.Paul says while we were still in our sins.
The French post-structuralist Jacques Derrida wrote much in life about the impossibility of the pure gift. For humans it is impossible--at least without the grace received by saints--for us to give without some notion of economy, e.g. some notion of tit for tat, but Jesus gave himself in full acceptance of our ingratitude. In fact this is what marks the Trinity who has no need of anything outside of Itself, yet out of love and desire to share Itself created all creation. Here is where of the notion of the Covenant comes into play. A covenant is as different from a contract as marriage is from prostitution. A contract is an agreement between person for the exchange of objects or services. A convent invokes God, assuming he is real (this is a big deal because if God really exists than Covenants have real authority), and is an exchange of persons. Covenants create family bonds. God who is family, in and of Himself, wants nothing more than to make us family too. The Trinity is everywhere the Old and New Covenants (translated into Latin as Testament) of Sacred Scripture. Properly understood the very fact we divide salvation history in Old and New "Testaments", itself is logical implication of the Trinity.
Why don't people better understand this? I don't know, but it is not the only thing people fail to realize about Christianity. Since my conversion I've started to call it the greatest open secret on the planet. The Apostles who comprised the Church from the beginning took pains to teach our faith in creeds. Yet very little of what the Church has consistently and continuously held as dogma, in this manner since the beginning, is realized by even the majority of professing Catholics. The Church certainly isn't attempting to hide any of it. In fact just the opposite. However, there certainly is an overwhelming large noise to signal ratio, as practically everything outside of the Church seemingly seeks to condemn and malign it. We are barraged continuously through the mass media with negative myths about Catholicism that undoubtedly has predisposed the world, generally speaking, to a equally negative bias on a deeply unconscious level. It is as if some force in the world wants to so muddy the water that even people who sincerely search for it have difficulty seeing the bottom.
Finally I would assert that whether this is strictly true or not--although if it is not strictly true than the whole concept of strict truth must be abandoned--it deserves the respect of any cultured and intelligent person. In fact if it is not true. and therefore not really a revealed knowledge then it paradoxically becomes even more worthy of respect from a materialist perspective. Since it then world be a work of sheer human genius.
That is the irony here: There is no case that can be made for holding such ideas in contempt, except perhaps if the contemptuous persons secretly fears it is true. he contempt just doesn't make any sense if one truly believed that this was only a human artifact.