The Letters Æ Ø Å -and the IPA
Æ - Ø - Å
People are always confusing "letters" with "sounds" and it really muddies the waters. Your textbook falls into this trap. The list of words they give under "pronunciation practice1" on page 4 gives a list of words that have the letters æ, ø, and å in them, and the implication is that the letters have the same sound in all of these words, but actually the pronunciations of each letter can vary across words.
The IPA
I've put the symbols used in the IPA (International Phonetic Alphabet) below to differentiate the sounds that occur in different words. Even if you don't know the symbols, you can see which words have the same sounds and which ones have different sounds in them. (The slashes / / always indicate that the symbols between them represent symbols in the IPA and represent one (and only one) sound. If there are no slashes, then the symbols are letters used in spelling and may have a lot of sounds associated with them).
If an IPA vowel symbol is followed by a colon (:), it means that it is a long vowel. The underlined letters are the ones that are pronounced as the IPA sound on the left.
Click the blue arrow to hear the word
Æ
/ɛ/ englænder sekretær (Englishman, secretary)
/ɛ:/ lærer (teacher)
Ø
/ø/ København øl børn søster (Copenhagen, beer, children, sister)
/ø:/ Jørgen Køge (man's name - cognate of "George," name of city south of Copenhagen)
/œ/ tømrer (carpenter)
Å
/ʌ/ nå så (oh/well, so)
/ə/ også (also)
/ɔː/ både (both)
/ɒː/ går (walk(s)/go(es))
/ɒ/ år (year/years)