Well, good question! I wonder if we will get any answers from theists. Meanwhile, I'll try and answer it without it turning into a farce.
Firstly, according to Chalcedon and Nicea, there is only one godhead though there are three persons within that godhead. So, technically, the question is worse than you thought - god sacrificed himself to himself to persuade himself to forgive sin
However, we might reasonably ask if that is what god was up to.
The first place in which this comes in the biblical text is in Paul. He compares the first man, Adam who sinned with Jesus, another man who showed staying clear of sin could be done. In a theological idea knows as Recapitulation, Everyone born from Adam dies, but those born (again) of Jesus live. Jesus has put things right. Christians rejoice in this, naturally, without really thinking it through.
I would argue that rather than the vicarious death of Jesus, Paul is alluding to the idea of Jesus showing what can be done and providing men with another chance
but it is the gospels who come up with more of the theology we know and hate today - Jesus dying for the sins of all people.
Why I can't say, but the whole Jesus story in the NT seems to leave the writers with the problem of putting the theology over, or at least getting into a coherent form. Thus they draw on OT passages - anything really that can just about be linked to Jesus to try and explain what they want to explain. Thus they come across the idea of transferring sins to a goat that is driven off into the desert carrying the people's sins. This sounded a good concept and thus is was transferred to Jesus. The writers only knew forgiveness via a sacrifice, apparently, though many Psalms and Jeremiah say so. This is where the disputes over the nature of Jesus came in.
To die for all sins, Jesus would have to be god - a lesser being wouldn't cut it. Yet to show that living sin free was possible, Jesus had to be a man. Thus was born the problem of Jesus to natures that lasted some 400 years in being solved - quaintly be restating the question.
Anyway, if one sees the message to live a sin free life, then the man Jesus shows it to one and Jesus has no need to be god and monotheism is preserved. On the other hand, if one accepts the gospel ideas of sacrifice, then it is a man-made idea that Jesus has to be god and the problems of the Trinity resulting are enough to blow anyone's brains apart.