How does a calculator compute sine and cosine?

Calculators do not store every possible sine and cosine value. There are infinitely many inputs, so a lookup table alone would not work. The usual idea is: 1. convert the angle to radians, 2. reduce it to a smaller equivalent angle using periodicity, 3. approximate the trig function with a polynomial or a related numerical method, 4. round the result to the display precision. For example, near 0 we have the Taylor expansion $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots.$$ But real calculators use more optimized versions than a naive Taylor series. They choose approximations that are fast and have a guaranteed error bound. So the calculator is not "knowing" sine. It is computing a very accurate approximation using identities and numerical analysis.