I found out that the input sensitivity is 25mv/47kohms, input impedance is 600 ohms, and the connector on my tape recorder is female and takes a 6.3 mm plug. Are the beta 57A and SM58 likely to work with the tape recorder?
25mV equals -32 dBV. The average output level of the Beta 57A is -71 dBV, 39 dB less than required. The average output level of the SM58 is -74 dBV, 42 dBV less than required. So neither mic is a good match to the specifications of your tape recorder.
The use of an in-line transformer, like the Shure A95UF, will increase the signal level of either mic about 20 dB. This still leaves about 20 dB of gain to make up to equal the input sensitivity of -32 dBV. Using such a transformer may be a perfectly good solution. You won't know until you try it.
If the -32 dBV number is correct and the in-line transformer does not provide enough signal gain for acceptable operation, you will require a mic preamplifier for any type of professional microphone you choose. A reasonably priced preamp is the Audio Buddy from www.midiman.com
We hope this answers all your questions.