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:

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Now evaluate these derivatives:

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Finally, simplify the result a bit:

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

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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:

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This estimate is accurate to four decimal places.

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