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

To estimate the length θ of a circular arc of the unit circle, the seventeenth-century Dutch scientist Christian Huygens used the approximation θ ≈ (8b − a)/3, where a is the length of the chord A̅C̅ of angle θ and b is the length of the chord A̅ B̅ of angle θ/2 (Figure 12).

(a) Prove that a = 2 sin(θ/2) and b = 2 sin(θ/4), and show that the Huygens approximation amounts to the approximation

(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|θ|⁵.

Transcribed Image Text:

Ꮎ ; 22 ર 16 0 sin 3 4 2/3 sin 0 2