As other people have mentioned in the comments, the vowel in words like "rang" and "rank" is traditionally transcribed with the symbol /æ/, called "ash," which corresponds to the vowel phoneme in the word "ash" in a standard modern English accent. This is the transcription you will see in dictionaries. It will likely to continue to be used because it is easy to derive most of the regional variants from this phonemic representation.
However, there are many American English accents (my own among them) where the vowel phone in the words "rang/rank" sounds significantly different from the vowel phone in the word "ash". The phoneme /æ/ is especially prone to being "tensed" before nasals and voiced velar sounds, and /ŋ/ is both.
For some speakers, /æ/ is raised before any nasal to something like
[ẽə̯̃]. I think this is about how I pronounce the vowel in "rang/rank" (to me, it seems more or less the same as the vowel in "ran" or "ram," and about the same in quality as the sound in "rare" or "rail"). Actually, if I pronounce "rang" slowly I can hear that the vowel in "rang" ends on a higher quality than the vowel in "ram," but I still don't mentally recognize my "rang" vowel as a closing diphthong.
For other speakers, the following velar nasal causes a high off-glide that is prominent enough that they perceive the vowel in "rang" to be noticeably different from the vowels in "ran" and "ram". For some of these speakers, there may be some kind of merger or near-merger with the phoneme /eɪ/ as in "rate." This doesn't cause much disruption to the system of vowel contrasts in English because the sequene /eɪŋ/ does not exist otherwise. This is similar in some ways to the change of /ɪŋ/ to [ɪjŋ] or /iŋ/ that is observed in some American English speakers, although I don't know if these changes tend to occur in the same areas or not. (For more on "ing", see the following post and the various linked posts: Why is /ɪŋk/ used with "ink" words when the actual pronunciation is /ijŋk/?)
Another similar change may occur for some speakers before the /g/ vowel, where it does cause a merger of previously distinguishable sounds (although the merged sound is sometimes perceived, or at least described as being closer to /æ/ than to /eɪ/). This is mentioned in the following post: Pronunciation of vowel in vague as [æ] instead of [eɪ].