 In this video, we're going to apply the remainder estimation theorem to find the minimum degree of the Taylor polynomial about x equals 0 needed to calculate sine of 1 with an error of less than 0.0005. Let's first define our function as sine of x. What do we know about a potential value for m in the theorem? We know that any derivative of sine of x is among the following. Sine of x, negative sine of x, cosine of x, and negative cosine of x. Now the maximum value of any of these for any x is 1. So we have that the absolute value of the n plus first derivative of f must be less than or equal to 1. So in our theorem, let's let m be 1. As a result, we now have that the error bound is bounded above by 1 over n plus 1 factorial times the absolute value of x minus x naught to the n plus 1. Well, what is our interval of interest? Well, we're considering the Taylor polynomials centered at 0. So x naught is 0. Since we're investigating the error involved in approximating sine of 1, the other side of our interval is 1. So this value will be 1. This value will be 0. So we're interested in the error on the interval from 0 to 1, which means that we want to find n such that the error bound is bounded above by 1 over n plus 1 factorial times the absolute value of 1 minus 0 to the n plus 1, which is equal to 1 over n plus 1 factorial. And we want that error to be less than 0.0005. Another way to think about this is to identify a value of n such that n plus 1 factorial is greater than 20,000. Since n represents a degree of a polynomial, we will test only whole numbers for n. Pause this video and test some values out. We find that since 6 plus 1 factorial is equal to 5,040 and 7 plus 1 factorial is 40,320, we need at a minimum a seventh degree Taylor polynomial to approximate the actual value of sine of 1 with less than a 0.0005 error.