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Chapter 8: Problem 5

Calculate \(\pi\) by approximating the smallest positive zero of sin. (Either bisect intervals or use Newton's Method of Section 6.4.)

Short Answer

Expert verified
\(\pi\) can be computed by approximating the smallest positive zero of sin using the bisection method. The key is to iteratively narrow down the interval containing \(\pi\) until the desired accuracy is reached.

Step by step solution

01

Set the Interval

Choose an interval that we know includes the zero point of the sin function, i.e., an interval \(I = [a, b]\) with \(a<\pi<b\). A natural choice would be \([3, 4]\) as \(\pi\) falls between these two values.
02

Bisection Method

The bisection method is to find the midpoint \(c = (a+b)/2\) of the interval \([a, b]\). Compute \(sin(c)\), if \(sin(c)\) is close to zero (within a specified tolerance), then we believe \(\pi\) to be this value. If not, check if \(sin(c)\) is positive or negative. In the case of positive, replace \(b\) with \(c\). Otherwise, replace \(a\) with \(c\). This process keeps repeating until we find a satisfying approximation for \(\pi\).
03

Determine the Tolerance

Before starting the iteration process, we need to specify a tolerance level to determine when to stop iterating. Let's say the tolerance is \(0.0001\).
04

Implement the Iteration Process

Start the iteration by finding the midpoint \(c\), then calculate \(sin(c)\). If \(sin(c)\) is within the defined tolerance, we stop iterating and take \(c\) as the approximation of \(\pi\). Otherwise, we check the sign of \(sin(c)\) to decide whether to replace \(a\) or \(b\) and continue with the new interval. We stop once \(sin(c)\) is within the tolerance level, then we confirm \(c\) to be the approximation of \(\pi\).

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