“They let women do some things at NASA,” says Taraji P. Henson as mathematician Katherine G. Johnson in “Hidden Figures,” the new biopic about the black women who played key roles at the agency during the height of the space program in the 1960s.

“And it’s not because we wear skirts,” she adds. “It’s because we wear glasses.”

It’s a line delivered to a suitor (Mahershala Ali) who expresses surprise at the nature of Katherine’s job, and it’s the sort of carefully-crafted dialogue that’s perfect for advertising a tough-minded, girl-power aesthetic in a trailer. And, of course, you’ll find it in the “Hidden Figures” ads.

And yet it doesn’t matter that it plays like something no actual person might say; real people usually stumble for words rather than eloquently expressing themselves in such a fashion.

That’s because the movie is such a genuine, rousing crowd-pleaser — a winning tribute to women never properly saluted for their brilliance — that it’s easy to embrace a touch of Hollywoodization.

In addition to Johnson, who joins the Space Task Group headed by Al Harrison (Kevin Costner) in 1961, with the race on between the Americans and Soviets for that first and crucial manned launch, this adaptation of Margot Lee Shetterly’s book of the same name depicts mathematician Dorothy Vaughan (Octavia Spencer) as she works to get ahead of her competition by learning about the giant IBM computer that’s promised to change everything; and Mary Jackson (Janelle Monáe), who dreams of becoming the space program’s first black female engineer.

Director Theodore Melfi works in broad strokes, interweaving the stories of these women with that of the Mercury program at large — particularly the improbable efforts to pull off the recently-departed John Glenn’s iconic orbit — and the movie offers plenty in the way of entertainment value simply in the urgency with which it returns to the idealism and ambition of the period.

As a reminder of what can happen when Americans pull together behind a collective and ambitious cause, it’s welcome, a rejoinder to the great strife and division that permeates the national mood today.

But, it’s not dishonest. The film depicts NASA’s Langley Research Center as a progressive institution relative to the realities of Virginia circa 1961, but that only goes so far: the women are restricted to employment in one wing, only as “Women Computers”; they can’t share coffee pots or bathrooms; and their white superiors practice sneering condescension at best, and downright hostility at worst.

There’s one exception, though. It’s Costner’s Harrison, who understands that the mission to launch a human being into space transcends such earthbound discriminatory standards.

He gives Henson’s Johnson the chance to showcase her brilliance and his utterly natural performance adds further credence to the notion that the actor is never better than he is when playing men of this period (see “JFK” or “Thirteen Days,” as two other examples).

Spencer and Monáe are as dependable and engaging as ever, but “Hidden Figures” really belongs to Henson. It’s hard to translate genius in a way that transcends the cerebral, but she finds the key in an understated fashion that expresses both her character’s isolation on the job and her unwavering belief in her ability to do it.

The movie celebrates the perseverance and determination of these characters, and does so in an accessible fashion. When the music swells and the other characters look on in awe as Johnson concocts elaborate formulas on super-sized chalkboards, when rooms full of big shots pause to celebrate her intellect, it’s genuinely thrilling.