Problem 54
Question
Improving approximations of \(\pi\) \begin{equation} \begin{array}{l}{\text { a. Let } P \text { be an approximation of } \pi \text { accurate to } n \text { decimals. }} \\ {\text { Show that } P+\sin P \text { gives an approximation correct to } 3 n} \\ {\text { decimals. (Hint: Let } P=\pi+x . )} \\ {\text { b. Try it with a calculator. }}\end{array} \end{equation}
Step-by-Step Solution
Verified Answer
The approximation \( P + \sin P \) improves the decimal accuracy of \( P \) to three times the original precision.
1Step 1: Analyze the Given Approximation
Let \( P \) be an approximation of \( \pi \) that is accurate to \( n \) decimals. This means that \( P = \pi + x \) where \( |x| < 10^{-n} \). Our goal is to show that \( P + \sin P \) gives an approximation of \( \pi \) correct to \( 3n \) decimals.
2Step 2: Use Small Angle Approximation for Sine
Assume that \( P = \pi + x \), then \( \sin P = \sin(\pi + x) \). Using the identity for sine around \( \pi \), we have \( \sin(\pi + x) = \sin(\pi)\cos(x) + \cos(\pi)\sin(x) = -\sin x \). For small \( x \), \( \sin x \approx x \). So, \( \sin(\pi + x) \approx -x \).
3Step 3: Compute New Approximation
Substitute \( -x \) into \( P + \sin P \): \[ P + \sin P = (\pi + x) - x = \pi. \] This shows that adding \( \sin P \) to \( P \) cancels the error term \( x \), theoretically resulting in \( \pi \).
4Step 4: Error Analysis for Precision
Given \( P = \pi + x \) where \( |x| < 10^{-n} \), when \( \sin P = -x \), the error in \( P + \sin P \) is reduced to \( |x - x^3/6 + \ldots| \). Since \( x^3/6 < (10^{-n})^3/6 < 10^{-3n} \), the next leading error term starts around \( 10^{-3n} \), meaning it is accurate to \( 3n \) decimals.
5Step 5: Verify with Calculator
Use a calculator to approximate \( \pi \) to a given \( n \) decimal and compute \( P + \sin P \). For example, using \( P \) accurate to 2 decimals: if \( P = 3.14 \), then \( \sin(3.14) \approx 0.00159265 \). Thus, \( P + \sin P \equals 3.14159265 \).
Key Concepts
Small Angle ApproximationDecimal AccuracyError AnalysisSine Function Approximation
Small Angle Approximation
Small angle approximation is a simplification used in trigonometry when dealing with angles close to zero. For small values of an angle, the sine of the angle is approximately equal to the angle itself when measured in radians. This is because for small angles, the arc length approaches the length of the corresponding chord, making \(\sin x \approx x\).
In this exercise, we're considering the angle \(x\) to be a small deviation from \(\pi\). When applying the small angle approximation to \(\sin(\pi + x)\), it simplifies to almost \(-x\) because \(\sin(\pi) = 0\) and the remaining term is \(-\sin x\).
This is crucial because it enables us to correct any small error in our approximation \(P\). By understanding and using this approximation, we can tweak our calculations for better accuracy.
In this exercise, we're considering the angle \(x\) to be a small deviation from \(\pi\). When applying the small angle approximation to \(\sin(\pi + x)\), it simplifies to almost \(-x\) because \(\sin(\pi) = 0\) and the remaining term is \(-\sin x\).
This is crucial because it enables us to correct any small error in our approximation \(P\). By understanding and using this approximation, we can tweak our calculations for better accuracy.
Decimal Accuracy
When we talk about decimal accuracy, we are referring to how many digits in a number are correct. In this context, having an approximation accurate to \(n\) decimals means that \(P\) is close to \(\pi\) by less than \(10^{-n}\).
For instance, \(P = 3.14\) is an approximation of \(\pi\) with accuracy to 2 decimal places. However, if we adjust this approximation using the small angle correction \(P + \sin P\), we can improve it to much higher precision. This means the result is accurate to more decimal places than the original approximation, specifically \(3n\) decimals in this calculation.
This technique is powerful because it allows significant improvement in precision without needing highly complex calculations.
For instance, \(P = 3.14\) is an approximation of \(\pi\) with accuracy to 2 decimal places. However, if we adjust this approximation using the small angle correction \(P + \sin P\), we can improve it to much higher precision. This means the result is accurate to more decimal places than the original approximation, specifically \(3n\) decimals in this calculation.
This technique is powerful because it allows significant improvement in precision without needing highly complex calculations.
Error Analysis
Error analysis involves examining how errors propagate in calculations and how they affect the result. It helps us understand the reliability and precision of our approximations.
In the original solution, we see that the error when calculating \(P + \sin P\) can be reduced drastically using the small angle approximation. If \(x < 10^{-n}\), the small angle approximation changes the effective error to a cubic term \(x^3/6\), which decreases fast for small \(x\).
This cubic term becomes insignificant compared to \(x\) because it is diminished by a factor of three powers. Therefore, \(P + \sin P\) has an error that starts decreasing at \(10^{-3n}\), rendering our new approximation accurate to \(3n\) decimal places. Understanding and managing these error terms is essential for improving the efficacy of mathematical approximations.
In the original solution, we see that the error when calculating \(P + \sin P\) can be reduced drastically using the small angle approximation. If \(x < 10^{-n}\), the small angle approximation changes the effective error to a cubic term \(x^3/6\), which decreases fast for small \(x\).
This cubic term becomes insignificant compared to \(x\) because it is diminished by a factor of three powers. Therefore, \(P + \sin P\) has an error that starts decreasing at \(10^{-3n}\), rendering our new approximation accurate to \(3n\) decimal places. Understanding and managing these error terms is essential for improving the efficacy of mathematical approximations.
Sine Function Approximation
The sine function is central in many areas of mathematics and engineering. Understanding its behavior, especially in approximations, is vital.
In this context, the function \(\sin(x)\) is approximated for very small \(x\) values. Remember that \(\sin(0) = 0\), so for arguments close to zero, \(\sin x \approx x\) becomes a lifeline.
In our problem, since \(\sin(\pi + x) = -\sin x\), knowing that \(\sin x \approx x\) lets us substitute and calculate with precision.
In this context, the function \(\sin(x)\) is approximated for very small \(x\) values. Remember that \(\sin(0) = 0\), so for arguments close to zero, \(\sin x \approx x\) becomes a lifeline.
In our problem, since \(\sin(\pi + x) = -\sin x\), knowing that \(\sin x \approx x\) lets us substitute and calculate with precision.
- This substitution dramatically changes the precision of \(P + \sin P\).
- From a trigonometric standpoint, understanding sine behavior leads to very accurate approximations with minimal computational effort.
Other exercises in this chapter
Problem 54
Which series in Exercises \(53-76\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty}(\sqrt
View solution Problem 54
Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{\ln n}{\ln 2 n} $$
View solution Problem 54
How many terms of the Taylor series for \(\ln (1+x)\) should you add to be sure of calculating ln \((1.1)\) with an error of magnitude less than \(10^{-8} ?\) G
View solution Problem 55
Estimate the value of \(\sum_{n=1}^{\infty}\left(1 / n^{3}\right)\) to within 0.01 of its exact value.
View solution