To estimate the length of a circular arc of the unit circle, the seventeenth-century

Chapter 8, Problem 60

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To estimate the length of a circular arc of the unit circle, the seventeenth-century Dutch scientist Christian Huygens used the approximation (8b a)/3, where a is the length of the chord AC of angle and b is the length of the chord AB of angle /2 (Figure 14). (a) Prove that a = 2 sin(/2) and b = 2 sin(/4), and show that the Huygens approximation amounts to the approximation 16 3 sin 4 2 3 sin 2 (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 approximation is less than 0.00022|| 5.