Taylor polynomials. This is a simple operation on AVR let's say, and only a little more complicated on anything else (Z80?). AVR gets you an 8 bit multiply, so it takes a lot of accumulating, but that's understandable with ten digits or however many. The basic function is:
f(x) = (((A*x + B)*x + C)*x + D)*x + ...
where A, B, C, D, ..., are the usual polynomial coefficients, factored into the above running form. Use as many terms as are necessary for the desired accuracy. Since accuracy is fixed by the display's digits, this is easy to determine beforehand.
Notice that, for sin and cos, every other coefficient factor can be skipped, since only even or odd powers of x are necessary. Also, only values between
0 and pi/2 are necessary; the rest can be calculated from identities. The reduced range also simplifies the polynomial.
Constants, like e and pi, can of course be stored, though they can be calculated from power series (from the exp and atan functions, respectively).
Special attention might be paid to optimizing in BCD, since everything's going out to a decimal display anyway (unless you want to read floating point hex, which is certainly possible, but rather unconventional..).
As for integration, there are algorithms which compute them to arbitrary precision. No big deal here.
But really, I'd just buy a $10 calculator from the grocery store, because they've already done all that, and Idunno, it's probably all integrated into one ASIC anyway (probably programmed the same way though, no fancy math hardware like in a PC).
Tim