Problem: compute cos(5) to two significant digits after the decimal point. As always, there are many ways to skin the cat, but here is one simple approach: Step 1: set y = [2Pi - 5], then cos(5) = cos(2Pi - y) = cos(y). Now y ~ 1.2 > 1, so still not too good for straight Taylor series (around 0). Step 2: cos(y) = sin(Pi/2 - y). Note that Pi/2 -y ~ 0.29 < 1 and small enough for the Taylor Series to converge fast. Step 3: Now you can expand sin(Pi/2 -y) = (Pi/2 - y) - 1/6*(Pi/2 -y)^3 .... The next term is of order (Pi/2 -y)^5 ~ 0.3^5 ~ 10^-3, only affects the third digit after the decimal point, so OK not to include. If you were to keep only the 1st term, that would not solve the problem since the answer would not have the required accuracy.