The adjective here describes what kind of composer she is, so it agrees with "композитор" not with "она".
If the sentence was "она композитор и она очень известная" - "she is a composer and she is very famous" then the adjective would agree with the pronoun and would be feminine.
No, it doesn't match the spelling of the noun, it matches the grammatical gender of the noun.
композитор is a masculine noun so it requires masculine adjectives; мужчина is also a masculine noun so it also requires masculine adjectives. The fact that мужчина ends in -а is perhaps misleading to the learner but is not what decides whether the noun is masculine or feminine grammatically.
And just as a woman is described by a masculine noun композитор in this sentence, so a man could be describes by a feminine noun such as особа "person" -- for example, Иван известная особа "Ivan is a famous person" with feminine adjective ending.
Many feminine forms of profession nouns sound sort of non-serious... I don't know how to say this in English. Врач (masculine) = doctor. Врачиха sounds very colloquial and not respectful enough. I can say so about a female doctor who does not seem competent enough or is rude to me. If I talk about a female doctor whom I really respect, I'd only say "врач". This is just an example.
Hmmm, I think I understand. So is it like that the female ending "-иха" etc. also has the sense of "little/ lesser" or some such?
If so, can one use this ending to modify nouns intentionally? We can do this in Irish, for example "buachaill = a boy (Irish)" while "buachaillín ~= a 'little' boy (colloquial)". The 'proper' way say "a little boy" would be "buachaill beag".
As an aside, you may find it interesting that in Irish we also do not have an indefinite article, as you see from these examples. This makes the lack of one in Russian quite natural to me!
@DesmondAllen I'm not aware of "actress" being out of favor in American English. If it's widely considered derogatory to women, shouldn't there be a major movement to change the name of the Academy Award for Best Actress?
Speaking of movies, interesting how композитор resembles the English "compositor."
it's historical and educational (mostly I would say!!) .... because women were considered as if they had no places in professional parts in society in most parts of the world...it happened but it was very rare and usually not recorded (written). In Europe during WWII women were working because men were up front fighting and there was a shortage of men...then when the war was over, most European governments asked women to go back where they were...at home...but...they would not! Anyway...I have here an ironic example is that you can say female doctor in English but in French you can say doctoress which the ---"ess" is derived from English like manageress (which is recognised in English but not the word doctoress!) ! another example: "écrivaine"-female writer in French is now in most dictionaries but made ridiculed sound wise when used... It will take few centuries to be accepted...maybe sooner if optimistic and less cynic! :)
It marks me wrong for using "the" instead of "a" -- so what is the difference? I would simply report but in this case I don't know, so I will ask and if answered it may be useful information for others. How would you express the definite versus the indefinite in this particular sentence?
In English, композитор would be termed a "predicate nominative" = a word which further describes the subject (Она) of the sentence. In English, predicate nominatives as usually found on the opposite side of the verb "to be". Since there is no present tense use of "to be" in Russian in sentences like this, I'm curious as to what part of speech words like композитор are determined to be in Russian. Here, it seems to be in nominative case, so it makes me think "predicate nominate" - but is that even a part of speech in Russian?