Finite rhymes with 'my time'. Infinite rhymes with 'minute' (as in seconds, minutes, hours and not minute, micro, small).
Why are i's in finite and infinite pronounced differently?
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Finite is pronounced ['faɪnaɪt] while infinite is pronounced [ˈɪnfɪnɪt]. So why is the vowel in the first syllable of 'finite' different from the vowel in the second syllable of 'infinite'?
OK, it's not because of Trisyllabic Laxing, I made an erroneous assumption which I deeply regret.
The answer is actually simple. As Luigi Burzio explains in Principles of English stress, the reason boils down to English stress patterns.
The diphthong [aɪ] (as in bite) almost never occurs in unstressed syllable. The diphthong [aɪ] has a systematic relationship with the short vowel [ɪ]. This relationship is also reflected in Trisyllabic Laxing; divine-divinity, derive-derivative and in drive-driven etc.
From this relationship, we can infer a general rule of thumb that [aɪ] will only occur in stressed syllables and when that syllable gets unstressed, [aɪ] will shorten to [ɪ].
Now when you prepend the prefix in- to finite, the primary stress moves to the prefix in- because it's a stress-bearing affix.
Another example would be cycle-bicycle:
There are exceptions, however. Luigi Burzio has explained all the rules and exceptions thoroughly in his book. One of the many exceptions is the prefix un- which doesn't take primary stress, for instance, unab[eɪ]ted.
Latinate words in English tend to be stressed on the penultimate (second last) syllable unless that syllable is short, in which case the primary stress falls on the antepenult (third last syllable) but not on preantepenult (fourth last) as far as I know. That's why impossible, implausible, incredible etc., are stressed on the antepenult.