How Do We Know The Trinity Is True?

There isn't a passage in the Bible that specifically says anything like "Oh by the way, The Father, The Son a.k.a. The Word a.k.a. Jesus, and The Holy Spirit are three persons who make one being - Galapagos 13:69". In fact the word 'trinity' doesn't appear anywhere. But that doesn't mean that this doctrine isn't a clear truth of Christianity. The word 'trinity' perhaps came later to be a simple description of a recurring theme throughout both the Old and New Testament.
But still there are a large number of Christians who think that God and Jesus were separate beings, and have no idea what the Holy Spirit is about.

May the grace of the Lord Jesus Christ, and the love of God, and the fellowship of the Holy Spirit be with you all. -- 2 Corinthians 13:14

Here we'll take a look at some of the several reasons that the Trinity is pretty much indisputable.

Firstly, without even looking at specific Scripture, let's take a moment to think about the atonement from the crucifixion and resurrection of Christ.
Basically, any human that sins must die. Sin can not enter heaven, so a human carrying it will not be able to go to heaven. That means everyone. Every human has a debt to pay. They owe a life for their sins. Potentially, one sinless human could pay the debt for them. If the sins of one human create a debt of one life, then the life of a sinless person could balance it out. It would have to be a sinless person, because if the second person had their own sins, they would have their own debt.
So given one sinful human, and one sinless one, either one could die for to pay the debt. One for one.
But if we have two sinful humans, and one sinless one, and the sinless one offers his life to save another, he can only pay for one. So we're still left with one debt not paid off.
So if Jesus was just a man who lived a perfect life, surely he could only give his life to save one other person. Maybe his favourite disciple.
But if Jesus was God, that means he has an infinite, eternal life, which is worth a whole lot more than one human life. His death, which cuts off eternity, is an easy trade off for however many humans (with an average of around a 70 year life span) have ever lived, with change to spare.
So Jesus in his infinite nature as God, having lived a sinless life is the only one who can afford to pay the debt of humanity.
Of course, everyone still dies, so it might seem confusing that Jesus died to save us all from dying, when we all still die. What Jesus' death saves us from is sometimes referred to as 'the second death'. So when our bodies die, our spirit can live on, until it is given a new body for the New Heaven in the future. But without Jesus' sacrifice, we would have no right to be given that new body and be brought back from death. The second death as described in Revelation is the destruction of the souls who are still carrying their own sin.

I simply can not see how Jesus being only a man and not the incarnation of God could work in this scenario.

No man can by any means redeem his brother or give to God a ransom for him, for the redemption of his soul is costly, and he should cease trying forever, that he should live on eternally, that he should not undergo decay. -- Psalm 49:7-9

One more point that shows that the Trinity is a necessary aspect of God without diving into Scripture. God is defined as the embodiment of love. (1 John 4:8 -- Anyone who does not love does not know God, because God is love.) God is also described as having aseity, which means he exists fully without needing anything. (Acts 17:25 -- And he is not served by human hands, as if he needed anything.) But how could God be capable of loving, sharing love, and receiving love if he was alone? He would need someone else. But seeing as God does not need anyone else, and can embody love whether or not humans exist, it must be that the shared love is contained within himself, in his multiple persons. If he was alone, he would not be sufficient to embody love, but as three in one, he can be just that. A God who is not self sufficient, and needs something or someone else, is not the God Christianity describes. The Trinitarian God is sufficient, but a unitarian God is not.

But let's go on and look at the Scriptural support for the Trinity.

Go therefore and make disciples of all nations, baptising them in the name of the Father and of the Son and of the Holy Spirit. -- Matthew 28:19 

It's quite straight forward for everyone to link the normal concept of God with the part of the Trinity referred to as The Father. There are several passages that say things like "yet for us there is one God, the Father" (1 Corinthians 8:6). You don't need much more than that. This being is described among other things as being transcendent of Heaven and Earth, being perfect goodness, being the embodiment of love, being the foundation of logic, being the creator of all things, being all knowing, being self existent and sufficient, existing eternally, being unchangeable, not needing anything. God has several qualities that are unique to him, and no lesser being could possibly have them. Yet, both the Son, and The Holy Spirit are described with the same qualities. The Bible is explicit that only one God exists, (e.g. Isaiah 46:9 -- For I am God, and there is none else; I am God, and there is none like me.) so the idea that these are three gods is completely out of the question. Both Jesus and Paul affirm that there is only one God (Mark 12:29 -- Jesus answered, “The most important is, ‘Hear, O Israel: The Lord our God, the Lord is one. Ephesians 4:6 -- one God and Father of all, who is over all and through all and in all.).
God himself makes it clear that he has no equal: Isaiah 42:8 -- I am the Lord; that is my name; my glory I give to no other. Isaiah 48:11 -- How can I let myself be defamed? I will not yield my glory to another.
Meanwhile, right at the beginning of the Bible God says in Genesis 1:26 "Let us make man in our image, after our likeness." Notice the 'us' and the 'our'. He's talking in plural. You might suggest that he's talking to angels, but then humans aren't made in the image of angels: Genesis 1:27 -- So God created man in his own image, in the image of God he created him.
So given that there is only one God, who has qualities that can't be shared, let's take a look at how Jesus and the Holy Spirit measure up and whether or not they might count in that 'us'.

Jesus - The Son, The Word, The Logos, The Lamb, The Angel of the Lord

Jesus was put to death. But if he never did anything wrong, why was he given the death sentence? Who would do that?
The reason for Jesus' execution was under the charge of blasphemy. The Sanhedrin were extremely unimpressed by this street preacher who was actually claiming to be God. It happens several times.

When Jesus saw their faith, he said to the paralysed man, “Son, your sins are forgiven.” Now some teachers of the law were sitting there, thinking to themselves, “Why does this fellow talk like that? He’s blaspheming! Who can forgive sins but God alone?” -- Mark 2:5-7

“I and the Father are one.”
Again his Jewish opponents picked up stones to stone him, but Jesus said to them, “I have shown you many good works from the Father. For which of these do you stone me?”
“We are not stoning you for any good work,” they replied, “but for blasphemy, because you, a mere man, claim to be God.” -- John 10:30-33

Anyone who has seen me has seen the Father. -- John 14:9

Believe me that I am in the Father and the Father is in me, or else believe on account of the works themselves. -- John 14:11

I am no longer in the world; and yet they themselves are in the world, and I come to You Holy Father, keep them in Your name, the name which You have given Me, that they may be one even as We are. -- John 17:11

“Very truly I tell you,” Jesus answered, “before Abraham was born, I AM!” At this, they picked up stones to stone him -- John 8:58-59

In these passages, Jesus says or does something that is universally known to be something that is only possible for God. John 8:58-59 there is a reference to Exodus 3:14 where God as a burning bush speaks to Moses, calling himself 'I AM'.

I am one who testifies for myself; my other witness is the Father, who sent me. -- John 8:18

In this passage, Jesus refers to the Jewish practice of having at least two witnesses in court, but suggests that he by himself counts as that many.

It's quite clear that based on Jesus' own claims to be God that his disciples came to believe it too.

For in Christ all the fullness of the Deity lives in bodily form -- Colossians 2:9

Christ Jesus: Who, being in very nature God, did not consider equality with God something to be used to his own advantage -- Philippians 2:5-6

Great indeed, we confess, is the mystery of godliness: He was manifested in the flesh, vindicated by the Spirit, seen by angels, proclaimed among the nations, believed on in the world, taken up in glory. -- 1 Timothy 3:16

Christ, who is God over all, forever praised! Amen. -- Romans 9:5

our great God and Saviour, Jesus Christ -- Titus 2:13

Be shepherds of the church of God, which he bought with his own blood. -- Acts 20:28

 “The virgin will conceive and give birth to a son, and they will call him Immanuel” (which means “God with us”). -- Matthew 1:23

"In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God. All things came into being through Him, and not even one thing that was created came into being without Him . . . And the Word became flesh, and tabernacled among us" -- John 1:1-3, 14

No one has ever seen God, but the one and only Son, who is himself God and is in closest relationship with the Father, has made him known. -- John 1:18

I have spoken to you of earthly things and you do not believe; how then will you believe if I speak of heavenly things? No one has ever gone into heaven except the one who came from heaven—the Son of Man. -- John 3:12-13

"Thomas answered and said to Him, "My Lord and my God!" -- John 20:28

Everyone who goes on ahead and does not abide in the teaching of Christ, does not have God. Whoever abides in the teaching has both the Father and the Son. -- 2 John 9

Now I want to remind you, although you once fully knew it, that Jesus, who saved a people out of the land of Egypt -- Jude 5

the righteousness of our God and Saviour Jesus Christ -- 2 Peter 1:1

And if the disciples calling him God isn't enough, the Father himself calls Jesus God.

But about the Son he says, "Your throne, O God, will last for ever and ever -- Hebrews 1:8

It also becomes clear that Jesus was sent by God from Heaven rather than being picked from humans born on Earth.
Jesus apparently first shows up in the New Testament, but actually he has some less obvious appearances throughout the Old too. We know that Jesus is God's incarnation on Earth. In Genesis, he appears and speaks to Hagar.

The angel of the Lord also said to her:

“You are now pregnant
    and you will give birth to a son.
You shall name him Ishmael,
    for the Lord has heard of your misery. 

 He will be a wild donkey of a man;
    his hand will be against everyone
    and everyone’s hand against him,
and he will live in hostility
    toward all his brothers.”

 She gave this name to the Lord who spoke to her: “You are the God who sees me,” for she said, “I have now seen the One who sees me.” -- Genesis 16:11-13

He is referred to as the angel of the Lord, but clearly shares the properties of being God and being in physical form. Perhaps at this stage he is not fully human, but he does have that important role of being a relateable bridge between humanity and divinity. There are other occasions too when God clearly appeared in human form.

So Jacob was left alone, and a man wrestled with him till daybreak. When the man saw that he could not overpower him, he touched the socket of Jacob’s hip so that his hip was wrenched as he wrestled with the man. Then the man said, “Let me go, for it is daybreak.”
But Jacob replied, “I will not let you go unless you bless me.”
The man asked him, “What is your name?”
“Jacob,” he answered.
Then the man said, “Your name will no longer be Jacob, but Israel, because you have struggled with God and with humans and have overcome.”
Jacob said, “Please tell me your name.”
But he replied, “Why do you ask my name?” Then he blessed him there.
So Jacob called the place Peniel^, saying, “It is because I saw God face to face, and yet my life was spared.” -- Genesis 32:24-30

The Lord appeared to Abraham near the great trees of Mamre while he was sitting at the entrance to his tent in the heat of the day. Abraham looked up and saw three men standing nearby. When he saw them, he hurried from the entrance of his tent to meet them and bowed low to the ground.
He said, “If I have found favor in your eyes, my lord, do not pass your servant by. -- Genesis 18:1-3

When the angel of the Lord did not show himself again to Manoah and his wife, Manoah realized that it was the angel of the Lord.
“We are doomed to die!” he said to his wife. “We have seen God!” -- Judges 13:21-22

He's often referred to as LORD, unmistakeably as God, which goes to prove that it is not out of the question in Jewish belief for Yahweh to come to Earth and communicate to us as a physical being. But the angel of the LORD is not always a physical being.

The angel of the LORD called to Abraham a second time from heaven -- Genesis 22:15

To think God can only be spiritual or physical at one time is called modalism and is not Biblically sound. God can be both.

Back to Jesus specifically, a prophecy in Isaiah spells out that the Messiah will be called God. In Jewish tradition names carried meaning and it's clear that the names given to the Messiah are a mixture of human and divine descriptions. 

For to us a child is born, to us a son is given; and the government shall be upon his shoulder, and his name shall be called Wonderful Counsellor, Mighty God, Everlasting Father, Prince of Peace. -- Isaiah 9:6

We've all heard Jesus referred to as that last one, if not the rest. As well as that, the passage clearly shows that a son who will be born is also the 'everlasting Father'. That's either a ridiculous contradiction, or a solid sign that Jesus and God are one being.

So Jesus claims to be God, and his followers believe that he is God, and there is precedence for God to take human form. But does Jesus share attributes with God? We've already seen a couple.

• God's chosen name: God specifically chose a name for his son which tells people what's what. Although in the west we call him 'Jesus', his actual given name was 'Yeshua', or the Greek form 'Joshua', which means "the LORD saves". Whenever you see 'Lord' in all caps like that, it refers to the Tetragrammaton, or God's own holy name. It's usually written as 'YHWH', or 'Yahweh', while again in the west, we make it our own and call it 'Jehovah'. So since in Hosea 13:4, God says he is the only saviour, yet Jesus is widely known to be our saviour, and this name 'Jesus' refers to both of them being saviours, it seems they must both be one and the same. Matthew 1:21 -- She will give birth to a son, and you are to give him the name Jesus, because he will save his people from their sins.
• Forgives sins: In Mark 2:5-7, Jesus forgives sins, something only God can do.
• Existed Eternally: John 1 and 3 tells us that Jesus was with one with God for all eternity and this is repeated elsewhere too. John 17:5 -- And now, Father, glorify me in your presence with the glory I had with you before the world began. In this passage the word 'world' is the Greek word 'kosmon' which refers to the system of the universe (sometimes physical, sometimes spiritual) rather than the Earth alone.
• He created things and is not a created being: John 1 also tells us that creation was through Jesus. 1:3 reads "All things were made through him, and without him was not any thing made that was made." This clearly says that all things that were made, were made through him. If he were a created (made) being, that verse would not make sense. Revelation 5:13 puts Jesus (the lamb) on equal placing with God on the throne above heaven. Hebrews 1:2 --  but in these last days he has spoken to us by his Son, whom he appointed heir of all things, and through whom also he made the universe.
• Sustains all things: Hebrews 1:3 -- The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful word.
• Authority over all creation: Matthew 28:18 -- Then Jesus came to them and said, “All authority in heaven and on earth has been given to me. 
• Lord of lords: Revelation 17:14 -- They will wage war against the Lamb, but the Lamb will triumph over them because he is Lord of lords and King of kings 
• Shares his name with God: Notice, 'the name above every name'. There is no exception given for God's own holy name, so it must be his own name that is given. Philippians 2:9-11 -- Therefore God exalted him to the highest place and gave him the name that is above every name, that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, and every tongue acknowledge that Jesus Christ is Lord, to the glory of God the Father. Similar to the first point in this list is that God gives his own name to Jesus. It's often said that no one knows for certain how to pronounce 'YHWH'. But if we take Jesus' given name 'Yeshua', and maybe go halfway with the western 'Jehovah', we get somewhere interesting. The Hebrews had no 'j' sound, and would have pronounced 'Jehovah', as 'Yehovah'. But the Tetragrammaton there has a 'w' where we have a 'v', so swap that out we get 'Yehowah'. It might just be me, but 'Yehowah', and 'Yeshua' seem almost identical, with perhaps the difference being that 'Yeshua' refers to the Son, while 'Yehowah' is the full trinity.
• Owns what God owns: John 16:15 -- All that belongs to the Father is mine.
• Is to be honoured exactly like the Father: John 5:23 -- Whoever does not honour the Son does not honour the Father who sent him.
• Calls himself equal to God: John 5:18 -- This was why the Jews were seeking all the more to kill him, because not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal with God.
  It's quite common for people to think that seeing as one is the Father, and one is the Son, that they should be separate. How can someone give birth to themselves? But despite seeming like common sense to us, the fact that Jesus called God his Father, and was himself called the Son of God, this was clear to the Jews of the time that Jesus was claiming to be equal to God. And as no one is equal to God, then they must be the same being. For the Jews, to call yourself a son of someone, wasn't always necessarily a biological thing. It was often used symbolically to mean you were the same type of person. A common phrase might be 'son of the prophets', which would mean the person being referenced was a prophet or preacher of some kind. So when Jesus claimed to be 'The Son of God', rather than 'a son of God', the cultural shall-we-say slang of the time made it clear that he was calling himself God.
  Many of his followers referred to him by that title (Mark 1:1, Mark 15:39, Luke 1:35, Luke 9:20, John 1:49), and even Satan and the demons(Matthew 8:28-29, Luke 4:3, Luke 4:41), but he never rejected it.
• Word to be kept: John 14:23 -- Jesus answered him, "If anyone loves me, he will keep my word, and my Father will love him, and we will come to him and make our home with him."
• All knowing: John 16:30 -- Now we can see that you know all things and that you do not even need to have anyone ask you questions. This makes us believe that you came from God.
• Knows human hearts and minds: John 2:25 -- He did not need any testimony about mankind, for he knew what was in each person. Luke 6:8 -- But He knew their thoughts.
• Never changes: Hebrews 13:8 -- Jesus Christ is the same yesterday and today and forever.
• Holy: Luke 1:35 --  The angel answered, “The Holy Spirit will come on you, and the power of the Most High will overshadow you. So the holy one to be born will be called the Son of God.
• Performed miracles and healings:  The Gospels are full of examples of Jesus using his ability to heal the sick, wounded, blind, and even the dying and dead. He also has control over the elements as shown by his walking on water, calming a storm, multiplying food, and changing water into wine. He doesn't even need to be in the same room to perform his miracles. Matthew 8:7-10,13 -- And he said to him, “I will come and heal him.” But the centurion replied, “Lord, I am not worthy to have you come under my roof, but only say the word, and my servant will be healed. For I too am a man under authority, with soldiers under me. And I say to one, ‘Go’, and he goes, and to another, ‘Come’, and he comes, and to my servant, ‘Do this’, and he does it.” When Jesus heard this, he marvelled...  And to the centurion Jesus said, “Go; let it be done for you as you have believed.” And the servant was healed at that very moment.
• A teacher: John 13:13 -- “You call me ‘Teacher’ and ‘Lord,’ and rightly so, for that is what I am.”
• Reveals Heaven's secrets:  In Matthew 13:10-17, Jesus explains that his parables explain the mysteries of Heaven. Galatians 1:12 -- I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ.
• Accepted worship: Even though in Matthew 4:10 Jesus says "You shall worship the Lord your God, and serve Him only.", there are several occasions when Jesus is worshipped by his followers (e.g. Matthew 2:11, 9:18, 14:33, 15:25, 28:9, John 9:38). Only God is worthy of worship. Men (Acts 10:25-26) and angels (Revelation 19:10) are not to be worshipped.
• Can be prayed to: Stephen prays to Jesus in his last moments in Acts 7:59, while Paul encourages prayer to Jesus in Corinthians 1:1-2, and Romans 10:13-14.
• Baptism in his name: Matthew 28:19 -- Go therefore and make disciples of all nations, baptising them in the name of the Father and of the Son and of the Holy Spirit.
• Will judge humanity:  2 Corinthians 5:10 -- For we must all appear before the judgement seat of Christ.
• Our advocate: 1 John 2:1 -- My dear children, I write this to you so that you will not sin. But if anybody does sin, we have an advocate with the Father--Jesus Christ, the Righteous One.
  
• Refers to himself as the first and the last: Revelation 1:17-18 -- "Do not be afraid; I am the first and the last, and the living One; and I was dead, and behold, I am alive forevermore, and I have the keys of death and of Hades." which is reminiscent of Isaiah 44:6 "Thus says the Lord, the King of Israel and his Redeemer, the Lord of hosts: ‘I am the first and I am the last, and there is no God besides Me." 
• Refers to himself as the Almighty: Just in case you weren't sure... Revelation 1:8 -- “I am the Alpha and the Omega,” says the Lord God, “who is, and who was, and who is to come, the Almighty.” This passage comes right after John says that Jesus is coming to give his revelation, and is clearly him introducing himself. Although it says 'Lord God' and not Jesus in this specific verse, the context of the previous verses makes it very clear. If not clear enough the word 'Lord' is translated from the Greek 'kyrios' (κύριος) which in the Bible is always used in reference to Jesus. When 'Lord' is applied to God the Father or the Tetragrammaton the Greek word used is 'theos' (Θεός).
  Knowing this distinction between 'kyrios' and 'theos' an interesting thing happens in Mark 12:30-31 -- And you shall love the Lord your God with all your heart and with all your soul and with all your mind and with all your strength.’ The second is this: ‘You shall love your neighbour as yourself.’ There is no other commandment greater than these.”These are the two greatest commandments for Christians. The first is to love the 'Lord your God'. Here Jesus says 'kyrion ton theon'. He mixes the two words and uses them to refer to God, basically saying you should love both him and the Father as one entity.
• Resurrects himself: Many verses will tell you that God raised Jesus from the dead. (Acts 10:40 -- God raised Him up on the third day and granted that He become visible). However, in John 2:18-22, Jesus explains that he will raise himself: So the Jews said to him, “What sign do you show us for doing these things?” Jesus answered them, “Destroy this temple, and in three days I will raise it up.” The Jews then said, “It has taken forty-six years to build this temple, and will you raise it up in three days?” But he was speaking about the temple of his body. When therefore he was raised from the dead, his disciples remembered that he had said this, and they believed the Scripture and the word that Jesus had spoken.

It seems pretty indisputable that the Bible teaches that Jesus and God are one and the same. The idea that Jesus is not a divine being made flesh is a serious sin according to John:

For many deceivers have gone out into the world, those who do not confess the coming of Jesus Christ in the flesh. Such a one is the deceiver and the antichrist. -- 2 John 7

The Holy Spirit

The Holy Spirit is a more difficult one to get a grasp of. It's hard to understand who it is, what it does, how it functions. Some people like to think of it as God's feminine side (despite always being referred to as 'he'), seeing as both the Father and the Son are referred to as male (even though God has no gender when he isn't assuming a human form).

We know that he is the helper sent to us by God. He is the part of God that lives inside each of us. He teaches, guides, helps, and comforts.

He too shares God's attributes. And we know that he is separate to both the Father and the Son because all three were present at Jesus' baptism by John (the Baptist).

And when Jesus was baptised, immediately he went up from the water, and behold, the heavens were opened to him, and he saw the Spirit of God descending like a dove and coming to rest on him; and behold, a voice from heaven said, “This is my beloved Son, with whom I am well pleased.” -- Matthew 3:16-17

A good verse that depicts the Trinity in action is this:

just as God said, "I WILL DWELL IN THEM AND WALK AMONG THEM; AND I WILL BE THEIR GOD, AND THEY SHALL BE MY PEOPLE. -- 2 Corinthians 6:16

This verse shows God dwelling within his followers (The Holy Spirit), walking among them (The Son), and being their God (The Father).

And let's see the matching qualities.

• Must not be blasphemed: Matthew 12:31 -- Therefore I tell you, every sin and blasphemy will be forgiven people, but the blasphemy against the Spirit will not be forgiven.
• Was present at Creation: Genesis 1:1-2 -- In the beginning, God created the heavens and the earth. The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. Note that in this passage 'God' in Hebrew is the word 'Elohim' which is a plural.
• Exists eternally: Hebrews 9:14 -- how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God.
• Called Lord: 2 Corinthians 3:17 -- Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. 
• Knows all God's secrets and understands the mind of God: 1 Corinthians 2: 10-11 -- these things God has revealed to us through the Spirit. For the Spirit searches everything, even the depths of God. For who knows a person's thoughts except the spirit of that person, which is in him? So also no one comprehends the thoughts of God except the Spirit of God.
• Omnipresent: Psalm 139:7-10 -- Where shall I go from your Spirit? Or where shall I flee from your presence^? If I ascend to heaven, you are there! If I make my bed in Sheol, you are there^! If I take the wings of the morning and dwell in the uttermost parts of the sea, even there your hand shall lead me, and your right hand shall hold me.
• A teacher on par with Jesus: John 14:26 -- But the Advocate, the Holy Spirit, whom the Father will send in My name, will teach you all things and will remind you of everything I have told you. In this passage again, we see that Jesus, the Father, and the Holy Spirit are separate persons.
• Provides revelations: Ephesians 3:5 -- which was not made known to men in other generations as it has now been revealed by the Spirit to God's holy apostles and prophets.
• Judges and convicts the world: John 16:7-8 -- Nevertheless, I tell you the truth: it is to your advantage that I go away, for if I do not go away, the Helper will not come to you. But if I go, I will send him to you. And when he comes, he will convict the world concerning sin and righteousness and judgement.
• Called advocate or helper by Jesus: John 14:15-17 -- “If you love me, keep my commands. And I will ask the Father, and he will give you another advocate to help you and be with you forever— the Spirit of truth.
• Baptism in his name: Matthew 28:19 -- Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.
• Loving: Romans 15:30 -- I appeal to you, brothers, by our Lord Jesus Christ and by the love of the Spirit.
• Referred to as God: Acts 5:35 -- Peter asked, “Ananias, why has Satan filled your heart so that you should lie to the Holy Spirit and keep back some of the money you got for the land? As long as it remained unsold, wasn’t it your own? And after it was sold, wasn’t the money at your disposal? So how could you have thought of doing what you did? You didn’t lie only to men, but also to God!”
• The Spirit of the Father: Matthew 10:20 -- For it is not you who speak, but it is the Spirit of your Father who speaks in you.
• Is both the Spirit of God and of Christ: Romans 8:9 -- You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him. 
• Is inside people and provides spiritual gifts: Exodus 31:3 -- I have filled him with the Spirit of God, with skill, ability and knowledge in all kinds of crafts. Judges 14:6 -- The Spirit of the LORD came upon him in power so that he tore the lion apart with his bare hands. Matthew 1:18 -- she was found to be with child through the Holy Spirit.
• with you and in you: Ephesians 4:6 says the Father is "over all and through all and in all", while John 14:17 says the Spirit "abides with you, and will be in you."

I struggle to see how anyone could say that the Trinity is not explicitly represented throughout the Scripture, even if it isn't explicitly explained in the Bible.

By the word of the LORD the heavens were made, And by the breath of His mouth all their host. -- Psalm 33:6

Certain passages of the Old Testament make more sense in light of the revelation of the trinity. This verse in Psalms refers to the word and the breath of the LORD. We know that Jesus is called the Word, and we also often see the Spirit referred to as breath.
Although Jesus and the Holy Spirit have certain characteristics explicitly referenced that don't match each other, there is a lot of overlap, and they both have qualities that only God could possibly have. So even though it's difficult to wrap your head around, the Father, Son, and Holy Spirit are all separate persons, yet they are all one being and one essence that make up the Trinitarian Godhead.
As well as the passages in the Old Testament that become clearer, there are a number of passages that seem to blur the lines between members of the Trinity. It's really unclear who is being referred to unless you see both persons as one and the same:

I charge you in the presence of God, who gives life to all things, and of Christ Jesus, who in his testimony before Pontius Pilate made the good confession, to keep the commandment unstained and free from reproach until the appearing of our Lord Jesus Christ, which he will display at the proper time—he who is the blessed and only Sovereign, the King of kings and Lord of lords, who alone has immortality, who dwells in unapproachable light, whom no one has ever seen or can see. To him be honour and eternal dominion. Amen. -- 1 Timothy 6:13-16

Each one is fully God. They are inseparable and they are not merely a part of God that can be removed. I've been using the word 'trinity' throughout so far, but it is not a Biblical word. A better more accurate word would be 'Godhead'. It might be possible that there are more than three in the Godhead. It doesn't really matter. Another better description of the Trinity would be tri-unity, which may have been the original term, but was squashed together. Tri-unity: three in unison.
Anyway, one last look at a vital passage that really drives the point home:

Go therefore and make disciples of all nations, baptising them in the name of the Father and of the Son and of the Holy Spirit. -- Matthew 28:19

In this all three members of the tri-unity, the Father, the Son, and Holy Spirit are to be referred to in the same name. The name is the Holy name of God ('YHWH' which some call Yahweh, or Jehovah, or Jah, or Adonai) . The word 'name' in the original Greek is 'onoma' (ὄνομα) which is singular, yet it is given to three persons. In the Old Testament Hebrew 'HaShem' which also means 'the name' is used to refer to God's holy name (e.g. Leviticus 24:11, Deuteronomy 28:58). 'the name' replaces 'YHWH' out of respect for God's holiest name, that no one is really sure how to pronounce. By simply saying 'the name' Jews know that refers to YHWH.
YHWH or Jehovah means "the self existent one" or as Moses heard it "I AM" or "I am that I am," or "I am the one who is". God has a number of other names that in many cases are more like titles which replace the Holy name. Adonai means 'one who rules' or 'lord'. Elohim (Genesis 14) means 'the strong God'. El Shaddai (Genesis 17) means 'Almighty God'. El Olam (Genesis 21) means 'Everlasting God'. 
This passage in Matthew refers to the name, so whichever one you want to use, it still comes to the same fact that it is referring to God - that the Father, the Son, and the Holy Spirit are all God.

By this you know the Spirit of God: every spirit that confesses that Jesus Christ has come in the flesh is from God, and every spirit that does not confess Jesus is not from God. This is the spirit of the antichrist, which you heard was coming and now is in the world already. -- 1 John 4:2-3