No one knows when cheetahs were first discovered because there is no record of that unique moment in the history of this much admired wild cat species. This is because it happened so long ago.
Cheetahs have been on the planet for far longer than “anatomically modern Homo sapiens1” (modern humans). It is said that cheetahs came about, in an evolutionary sense, around 6.7 million years ago when they diverged from the jaguarundi and puma (mountain lion). Modern humans emerged in Africa 200,000 years ago1. Of course early humans (hominids) could have discovered the cheetah.
I would have expected that the first modern humans would have regularly encountered cheetahs. If this is the case the cheetah was discovered by humans around 200,000 years ago. That is my best shot at an answer to the question. If someone can do better please leave a comment.
The first time a human named the cheetah scientifically (in Latin) was in 1776. A German zoologist, Schreber, named the cheetah Acinonyx jubatus. It took a long time for humans to get around to working out how the cheetah fitted in with the other cats in the family of cats (taxonomy).
The relationship between the sociable cheetah and humans has been fruitful for humans, particularly rich ones in the Middle East. This is perhaps a legacy of the ancient history of the region. Rich Arabs still like to turn cheetahs into pets but, recently, the law has been tightened up in UAE. This concept started at least 4,000 years ago. Cheetahs are enshrined on many Egyptian tombs. A cat that looks like a cheetah is on a Mesopotamian seal from the third millennium BC.
The first Indian potentate to hunt with cheetahs was Akbar the Great in 1555.