Final answer:
Katherine Johnson was the NASA mathematician who significantly contributed to the success of American spaceflights, including calculating the trajectory for the lunar landing of Apollo 11.
Step-by-step explanation:
The woman who played a pivotal role as a mathematician at NASA and helped send a successful manned mission into space is Katherine Johnson. Her expertise in orbital mechanics was critical to the success of the first American spaceflights, including Apollo 11's historic 1969 moon landing. Johnson's method of calculating launch windows by working backwards from the desired landing time was revolutionary for its time. She contributed significantly to NASA's success during her 33 years at the agency, co-writing 26 research reports. She also became widely recognized after the release of the book and subsequent film, Hidden Figures, which highlighted her contributions and those of other African American women at NASA.