Expressing the Function sin x as a Series

By Mark Zegarelli

If you want to find the approximate value of sin x, you can use a formula to express it as a series. This formula expresses the sine function as an alternating series:


To make sense of this formula, use expanded notation:


Notice that this is a power series. To get a quick sense of how it works, here’s how you can find the value of sin 0 by substituting 0 for x:


As you can see, the formula verifies what you already know: sin 0 = 0.

You can use this formula to approximate sin x for any value of x to as many decimal places as you like. For example, look what happens when you substitute 1 for x in the first four terms of the formula:


Note that the actual value of sin 1 to six decimal places is 0.841471, so this estimate is correct to five decimal places — not bad!

The table shows the value of sin 3 approximated to six terms. Note that the actual value of sin 3 is approximately 0.14112, so the six-term approximation is correct to three decimal places. Again, not bad. Though, this one wasn’t quite as good as the estimate for sin 1.


As a final example, the following table shows the value of sin 10 approximated out to eight terms. The true value of sin 10 is approximately –0.54402, so by any standard this is a poor estimate. Nevertheless, if you continue to generate terms, this estimate continues to get better and better, to any level of precision you like. If you doubt this, notice that after five terms, the approximations are beginning to get closer to the actual value.