# Expressing the Function cos *x* as a Series

If you want to find the approximate value of cos *x*, you start with a formula that expresses the value of sin *x* for all values of *x* as an infinite series. Differentiating both sides of this formula leads to a similar formula for cos *x**:*

Now evaluate these derivatives:

Finally, simplify the result a bit:

As you can see, the result is a power series. Here’s how you write it by using sigma notation:

To gain some confidence that this series really works as advertised, note that the substitution *x *= 0 provides the correct equation cos 0 = 1. Furthermore, substituting *x* = 1 into the first four terms gives you the following approximation:

This estimate is accurate to four decimal places.