What is the standard US pronunciation for words such as the following:
At least in my dialect of US English (Inland Northern), the following seem like close transcriptions:
However, in dictionaries that I am referencing (Japanese dictionaries of English), the vowel is given as /ae/, as in the following:
All of the above answers are confusing phonemic with phonetic transcriptions. Transcriptions like /bæŋk/ are broad phonemic representations. A transcription like [bɛɪŋk] (which is pretty standard for the US and about how I actually say it) is a narrow phonetic transcription (notice the brackets, not slashes). The Japanese dictionary of English is using a broad representation for two reasons:
Basically, before /ŋ/, Americans diphthongize and often raise front vowels. The words "kit" and "king" are broadly /kɪt/ and /kɪŋ/, but narrowly something like [kʰɪ̞ʔt] and [kʰɪiŋ]. In terms of just the vowels, the vowel in "kit" (and nearly everywhere else that "short i" occurs) is significantly lower than the cardinal IPA /ɪ/ sound. The vowel in "king" is a diphthong that begins higher than the vowel in "kit" and moves higher still.
BTW, the "Inland Northern" dialect is nowhere near Montana. It stretches along the major Rust Belt cities from approximately the Chicago metro area to the Rochester, NY metro area, covering areas that border Lake Michigan, Lake Erie and Lake Ontario, as well as most of the state of Michigan.
AmE pronunciation of Bank:
/baNGk//ˈbaŋk//bæŋk//bæŋk//bæŋk//beɪŋk/ or /bæɪŋk/Going by the above, the standard pronunciation appears to be closest to /bæŋk/. Wiktionary's is the only entry that notes something close to your provided pronunciation. I really can't hear anything similar on Forvo.
The pronunciation of bank is also discussed on phonetic-blog in great detail. One of the comments suggests that /æ/ raised to /eɪ/ before velars is an affectation peculiar to Montana, a location which fits in well with your "inland northern" geography.
