Related to this history is how calculators used CORDIC, COordinate Rotation DIgital Computer, to compute trigonometric functions in real time with minimal hardware. CORDIC was originally developed by Jack Volder working at Convair corporation in 1959.

Here is a nice explanation (PDF): https://www.qc.cuny.edu/Academics/Degrees/DMNS/Faculty%20Doc...

EDIT: in the HP-35 article it is mentioned in section III. The PDF here goes in to the math and implementation of CORDIC.