"The eagle is beautiful."
According to the information from the various Tips and Notes and if I understood them correctly, it is used in (at least) two cases:
- when the subject is indefinite (A book is good - ספר הוא טוב literally "(a) book he/it (is) good");
- when the predicative is nominal.
I am unsure whether the pronoun/copula is necessary when the subject is indefinite and the predicative definite (because it seems it would be unambiguous even without the copula)… For instance, I wonder whether דג החייה (rather than דג הוא החייה) could be correct to mean ‘a fish is the animal’. If anyone knows, you’re welcome to tell.
(The words from the version without copula will not be displayed correctly from right to left, sorry about this!)
I am not a native English speaker, so I don't know what "when the attribute is nominal" means, but here is when to use copula.
When you have a definite noun and an adjective, just like in this sentence, you don't need copula. However, sometimes when you have a longer subjet, it is possible to insert it, in order to avoid ambiguity.
When you have an indefinite noun and adjective, just like your first example, you need copula. פיל הוא גדול. An elephant is big. Otherwise, without a copula פיל גדול would mean "a big elephant".
When you equate two nouns, regardless if they are definite or indefinite, you need copula. So דג הוא חיה is correct. Not sure if I can think of an example right now when you would say that with only one side definite, as you wrote it.