91Ó°ÊÓ

The function \( \sin x \) is a fundamental concept in trigonometry, and its value oscillates between -1 and 1 as \( x \) changes. To approximate \( \sin x \) using a Maclaurin series, we expand the function around \( x = 0 \). A Maclaurin series is a type of Taylor series where the expansion point is zero.
The Maclaurin series for \( \sin x \) is an infinite series expressed as:

• \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \)

Each term involves an increasing power of \( x \) with factorials in the denominator to weigh them. These terms are constructed using the derivatives of \( \sin x \), which alternate between \( \cos x \) and \(-\sin x\).
Since \( \sin x \) is an odd function, the terms of the series alternate in sign, making it converge more smoothly. This alternation also helps improve the approximation more efficiently near \( x = 0 \).

When approximating functions with polynomials, the accuracy of the approximation is measured by the remainder term. The Lagrange remainder \( R_{n+1}(x) \) describes the difference between the actual value of a function and its polynomial approximation. For a Maclaurin series, this remainder is crucial in identifying how precise the polynomial is.
The formula for the Lagrange remainder in Maclaurin series is:

• \( R_{n+1}(x) = \frac{f^{(n+1)}(c)x^{n+1}}{(n+1)!} \)

Here, \( f^{(n+1)}(c) \) is the \((n+1)\)-th derivative of the function at some point \( c \) between 0 and \( x \). In the case of \( \sin x \), its derivatives cycle through \( \cos x, -\sin x, \) and \(-\cos x\), making the absolute maximum of \( f^{(n+1)}(c) \) always 1. This simplifies estimating the remainder, particularly when seeking high accuracy.
By ensuring the remainder \( R_{n+1}(x) \) is sufficiently small, such as less than 0.00005, we can decide on the appropriate degree of the polynomial for a desired precision.

Polynomial approximation allows us to estimate complex functions with simpler polynomial expressions. The Maclaurin polynomial is a tool for this, particularly useful for functions like \( \sin x \) where direct computation may be cumbersome for small angle approximations.
The order of the polynomial provides control over the approximation's accuracy:

• Higher order polynomials yield better approximation but require more computation.
• Odd orders are significant for \( \sin x \) since it is an odd function, ensuring that the approximation remains faithful to the function's nature.

To find how large \( n \) must be for \( \sin x \) to be accurate to within a certain tolerance, we assess \( R_{n+1}(x) \) across the desired interval. Higher order terms like \( \left(\frac{\pi}{2}\right)^{n+1} \) quickly diminish in scale when divided by a large factorial \((n+1)!\), offering increased accuracy.
By methodically solving the remainder inequality using trial and error with incremental odd \( n \), we achieve the balance between precision and computational ease. In this scenario, determining \( n = 11 \), allows \( \left|R_{12}(x)\right| \approx 0.000042 \), fulfilling our precision requirement.