Faxes rely on modems. modems rely on precise timing and aren't able to cope with any slippage.
There's actually a protocol, T.38, which is used on modern VoIP based systems to allow faxes to be transmitted. At each end it actually decodes the fax data and sends it as ordinary data packets. Implementations play some tricks on the fax modem at each end, in order to prevent it dropping out of sync if the data packets are delayed. You could communicate with one of these from phone, but you'd need a T.38 gateway a the other end.
More practical is to cut out the middleman and use twilio's fax gateway at $0.01/page (+ sip trunking cost, which should be similarly trivial): https://www.twilio.com/fax/pricing
There's actually a protocol, T.38, which is used on modern VoIP based systems to allow faxes to be transmitted. At each end it actually decodes the fax data and sends it as ordinary data packets. Implementations play some tricks on the fax modem at each end, in order to prevent it dropping out of sync if the data packets are delayed. You could communicate with one of these from phone, but you'd need a T.38 gateway a the other end.
More practical is to cut out the middleman and use twilio's fax gateway at $0.01/page (+ sip trunking cost, which should be similarly trivial): https://www.twilio.com/fax/pricing