discover may apply to something requiring exploration or investigation or to a chance encounter.
discovered the source of the river
ascertain implies effort to find the facts or the truth proceeding from awareness of ignorance or uncertainty.
attempts to ascertain the population of the region
determine emphasizes the intent to establish the facts definitely or precisely.
unable to determine the origin of the word
unearth implies bringing to light something forgotten or hidden.
unearth old records
learn may imply acquiring knowledge with little effort or conscious intention (as by simply being told) or it may imply study and practice.
I learned her name only today
learning Greek
Example Sentences
Several new species of plants have recently been discovered. It took her several weeks to discover the solution. The tests have discovered problems in the current design. She soon discovered what had been going on.
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Word History
Etymology
Middle English, from Anglo-French descoverir, descovrir, from Late Latin discooperire, from Latin dis- + cooperire to cover — more at cover