Richard Davies, 41, died of a single gunshot wound to the chest after firing at officers in St Neots, Cambridgeshire, in October 2015.
The father-of-three said he "wanted to end his life" after learning his marriage was over, the hearing was told.
The inquest in Peterborough is expected to last two-and-a-half weeks.
Mr Davies had been signed off work following difficulty with a colleague and had been prescribed anti-depressants, jurors heard.
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On the day of his death, he made several trips to a nearby shop to buy alcohol and had been carrying a knife.
He had a "long, emotional talk" with his wife Samantha when she arrived home and things seemed "fairly normal" before she left to see her sister.
When his children returned from school at around 19:00 BST, he sent two back out to buy wrapping paper and string, jurors at Peterborough Town Hall were told.
Mr Davies then tied up a third child, put duct tape over his mouth and locked him in a bedroom.
When the children returned, Mr Davies "brandished a kitchen knife; he was saying that he wanted to end his life," assistant coroner Nicholas Moss said.
"Mr Davies threatened to kill the children if they did not do what they were told, so Child A decided to allow Mr Davies to tie him up," he added.
The inquest then heard how the children managed to make 999 calls and alert their mother. All three escaped through upstairs windows.
Police marksmen attended within "four or five minutes", with Mr Davies assessed as being "emotionally or mentally distressed or disturbed".
He then fired a total of six shots from the house, the inquest heard.
Mr Moss said that Mr Davies shouted "f****** shoot me then you c****"', to which an officer replied: "We don't want to do that to anybody, Richard".
Mr Moss said one firearms officer - referred to as 'I7' - saw Mr Davies reappear at an upstairs window.
"He was concerned Mr Davies was going to fire again and he had no option but to fire," said Mr Moss.
He told jurors: "You will need to examine whether other options could or indeed should have been adopted."
The inquest continues.
An engineer shot dead by police had tied up his children and threatened to kill himself, an inquest has heard.