He divorced her a third time during a period of purity in which he had had intercourse with her

Praise be to Allah.

Firstly: 

Divorce (talaq) as prescribed in Islam is that in which a man divorces his wife during a period of purity in which he has not had intercourse with her. If he divorces her when she is menstruating or bleeding following childbirth (nifaas), or during a period of purity in which he has had intercourse with her, then this is an innovated divorce (talaq bid‘i) 

The fuqaha’ differed as to whether it counts as such. The majority are of the view that it does count as such, but some are of the view that it does not, because it is a haraam, innovated divorce, and Allah, may He be exalted, says (interpretation of the meaning): “O Prophet ! When you divorce women, divorce them at their Iddah (prescribed periods)” [at-Talaq 65:1]. What is meant is: when they are pure (not menstruating) and their husbands have not had intercourse with them. Among those who were of this opinion was Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him), who was followed by a number of scholars. 

It says in Fataawa al-Lajnah ad-Daa’imah (20/58): There are several types of innovated divorce (talaq bid‘i), such as when a man divorces his wife when she is menstruating or bleeding following childbirth (nifaas), or during a period of purity in which he has had intercourse with her. The correct view is that this does not count as such. End quote. 

Based on that, if you divorce your wife during a period of purity in which you have had intercourse with her, then it does not count as such according to the more correct view. 

Secondly: 

With regard to divorce issued in a state of anger, that is subject to further discussion. It has been discussed previously in the answer to question no. 45174 

And Allah knows best.