To estimate the length e of a circular arc of the unit circle, the seventeenth-century Dutch scientist Christian Huygens used the approx- imation 0 ~ (86 - a)/3, where a is the length of the chord AC of angle 0 and b is the length of the chord AB of angle 0/2 (Figure 12). (a) Prove that a = 2 sin(0/2) and b = 2 sin(0/4), and show that the Huygens' approximation amounts to the approximation 16 3 sin- sin NIS (b) Compute the fifth Maclaurin polynomial of the function on the right. (c) Use the Error Bound to show that the error in the Huygens' approxi- mation is less than 0.000221915. B h A FIGURE 12 Unit circle