“More and more people will understand that we’re not villains,” she said, noting that the exposure many veteran officers had to transgender people was limited to interactions on the job, negatively shading their impressions. “They see sex workers, they see drug abusers, they see people stealing to support their habit or to survive,” she added. “They translate their experiences with those people to all transgender people.”

Officer Brooke Bukowski of the New York Police Department got her start in what police officers call an “A-House,” one of the precincts in New York where crime and violence persists, the kind of place where young officers can earn their stripes. After the academy, she was assigned to the 75th Precinct, which covers the East New York section of Brooklyn and has consistently had one of the highest murder rates in the city.

Officer Bukowski, a United States Coast Guard veteran who was raised in rural Illinois, had started taking a low dose of hormones to transition before she joined the Police Department five years ago. She stopped when she entered the academy.

But once she graduated, she resumed taking the hormones. When she moved to the 70th Precinct, in Flatbush, her transition became more apparent. Her hair grew longer, and her body changed. While she was on patrol, people on the street would refer to her as Miss, and she remembered overhearing two men in custody discussing her. “They literally start arguing over what gender I am,” she said.

She reached a point where she was “male failing,” as she put it, a term in the transgender community meaning one could no longer pass as male. And after speaking with her superiors, she addressed roll call, explaining her transition.

“Listen guys, I still have the same work ethic,” Officer Bukowski, 33, recalled telling them. “I’m still going to back you up. It’s just a few physical changes here and there.”

Her colleagues have mostly been supportive and encouraging. But there have been obstacles, mostly bureaucratic encounters within the department. For instance, she had a hard time getting a new identification card. She was told she was not due for one yet, though her appearance had drastically changed. (She eventually got a new card.) And around the time that Ms. Jenner appeared on the cover of Vanity Fair, Officer Bukowski was working on a detail with an older officer who made disparaging remarks about transgender people, unaware that Officer Bukowski was herself transgender. She opted to quietly finish the shift without saying anything. “I had an hour left with the guy,” she said.