I think it all comes down to what the writer/speaker thinks about the reader/listener
ie. If it is new to the reader use 'a', otherwise use 'the'
as for your examples
So I just cannot see the difference between "this is a book I told you about yesterday" and "a woman who fell 10 meters from High Peak was lifted to safety by a helicopter."
Why can a definite article be used with "woman" for introducing for the first time and not with "book"?
If you told me about the book yesterday, then it wouldnt be new information so it would be: "this is the book i told you about yesterday", Using 'a' in this case almost implies that I forgot about it, and you are reminding me
For the second one, I have never heard of the[1] woman (in the article) before so you describe her as 'a' woman and not 'the' woman.
the[1] last examples:
It should be "the" according to the explanation people have given me, right? Unless there is more than one letter he sent to Burnham on June 20, which is unlikely.
This does read to me as if there is more than one or that he wrote so many letters over the years that they can only specify which one they are talking about by date.
Likewise, it should be "the" unless Burnham wrote several letters that contained a description of the meeting.
it is 'a' because he wrote only one letter. "Burnham wrote one letter to Olmsted...", if you said 'the' in this case, it would imply that I knew about the letter before you said the sentence
[1] I use 'the' because you know what I'm refering to here
My answer is semantic, not grammatical. The issue is one of meaning, not of rules of syntax.
If you intend that there is one unique correct answer then "the right answer" is the correct meaning since the noun "answer" is fully determined by the adjective "right".
Similarly, there are usually many incorrect answers: so "That's a wrong answer" is correct and meaningful. You are wrong to say that this is not grammatical. Similarly, you can use "the perfect car" to imply that the car is fully determined by its perfection, but you can also use "a perfect car" to imply that there are others that are equally good.
So in all the examples you give, both "a" and "the" are grammatically possible. A native speaker would intuitively choose an article according to the meaning that they want to give.
Generally, adjectives that are incomparable will be more determining. If you can't be "righter" or "less right", then a noun will be fully determined by the adjective and so the article chosen will be "the".
Best Answer
In choosing a or an, the only thing that matters is the sound following the article, whether it's an adjective, a noun, or what: a for consonant sounds, and an for vowel sounds. There aren't different rules for different parts of speech.
One of the confusing things about English is that the same combinations of letters can sometimes be pronounced in different ways depending on the context. (By the way, my phonetic spellings are based on my Northeastern US accent, but the important parts regarding this issue should be the same everywhere.)
And it happens that unisex is like unicorn, for the reasons that StoneyB mentions. So the appropriate article is a: a unisex _____ - because the y sound is considered a consonant sound.
Eu is another one that can be pronounced two ways:
but it's always a y sound when it's at the beginning of a word, so it's always a European vacation, a euphemistic explanation, and so forth.