EXAMPLE 2 (a) What is the maximum error possible in using the approximation sin xxx-xx 3! 5! when -0.8 ≤ x ≤ 0.8? Use this approximation to find sin 12° correct to six decimal places. (b) For what values of x is this approximation accurate to 0.00005? SOLUTION (a) Notice that the Maclaurin series sin x=x-x+xs - +... 3! 5! 7! is alternating for all nonzero values of x, and the successive terms decrease in size because |x| <1, so we can use the Alternating Series Estimation Theorem. The error in approximating sin x by the first three terms of its Maclaurin series is at most = - 1x7 7! If -0.8 ≤ x ≤ 0.8, then Ix ≤ 0.8, so the error is smaller than (0.8)7 (rounded to nine decimal places). To find sin 12° we first convert to radian measure: 12π sin 12° = sin 180 = sin Π 2 15 -(*)* 3 5 1 1 15 (rounded to eight decimal places). Thus, correct to six decimal places, sin 12° (b) The error will be smaller than 0.00005 if 1x17 <0.00005. Solving this inequality for x to three decimal places, we get 1x17< or x <(0.252) 1/7 So the given approximation is accurate to within 0.00005 when |x| < (rounded to two decimal places).