What arguments does a missionary use to try to convert people to Christianity?

Threats, coercion, sometimes violence.

Threats, intimidation and murder usually.

Because we have all sinned against a perfectly holy and righteous God, we all deserve eternal separation from God in hell (Rom. 6:23). However, God loves us so much that He sent His only Son Jesus to die on the cross as our Substitute so we don't have to go to hell. Christ died for our sins, was buried, and rose from the grave three days later (1 Cor. 15:3-4). He gives eternal life freely to anyone who simply believes in Him for it. No one can be saved by doing good works. Salvation is by grace alone through faith alone in Christ alone (Eph. 2:8-9).

"Verily, verily, I say unto you, He that believeth on me hath everlasting life" (John 6:47).

The same one others use to convert people to their beliefs.

FEAR is the most powerful tool ever devised.

~

out and out lies of course

What missionaries do, is merely present to them the Word of God, they decide if they want to believe it or not.

Misselling. Coercion. 

Fear. Testimonies that might be sincere or not. Coercion. Hard sell. Pick out the vulnerable and hurting, target them. Offering friendship and fellowship, then taking that away unless that person gives in, becomes a member. 

I was a missionary for the LDS church. I can tell you that arguments DO NOT convert. They only make the person arguing feel good about themselves.

Sincerely caring about and listening to another person builds trust. Then the person will be open to your thoughts. If what you have to offer is really true, and you can sincerely testify of it's truthfulness, the spirit will touch the person and he will feel of it's truthfulness. This is a powerful concept. The spirit goes deep inside and is long lasting. God wants us to know truth.

An example of testifing of truth might be:

"I know that prayer builds faith in God. It has changed my life and I know it will change yours". 

You can testify about all kinds of things such as eating good, applying gospel principles to family relationships, getting an education, staying morally clean, etc.

Don't try to give a fake testimony about something you do not know to be true or which is just mens interpretations of scripture. People sense the hypocricy. If you do not sincerely know what you are teaching, don't teach it. There is joy in teaching truth.